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Chapter 5
Conductors, Dielectrics and Capacitance
Current and Current Density
Current
IdQ
dtmotion of positive charges
Current Desnsity is a vector represented by J
I J S
I SJdot
d
Current and Current Density
IQ
tv S
x
t
taking the limit with respect to time
I v S vx vxrepresents the x component of the velocity
In terms of current density
Jx v vx in general J v v
Incremental of Charge
Q v S L
Example D5.1
Given the vecotr density J, calculate the current density at P
3 30
180 z 2
J z 10
2 z
4 cos 2
0
J z 180
9
0
Determine the total current flowing outward through the cirrcular band
3 0 2 2 z 2.8
I
2
2.8
z
0
2.
J z 3
0
0
d
d I 3.257 103
Continuity of Current
This equation indicates that the current diverging from a small volume per unit volume is equal to the time rate of decrease of charge per unit volume at every point
Total current through the closed surface
I SJdot
ddQi
dt
Using the divergence theorem
I SJdot
d vdel J( )
d
S vol
vdel J( )
dd
dtvv
d
vol vol
vdel J( )
d vdv
dt
d
vol vol
del J( ) vdv
dt v
del Jdv
dt
Conductors, Dielectrics, Semiconductors
The Energy Band Structure in Three Different Types of Materials at 0Ka) The conductor exhibits no energy gap between the valence and conduction bands.b) The Insulator shows a large energy gapc) The semiconductor has only a small energy gap
Table 2.1 Electrical Classification of Solid Materials
Materials Resistivity (-cm)Insulators 105 < < Semiconductors 10-3 < < 105
Conductors < 10-3
Conductors, Dielectrics, Semiconductors
Metallic Conductors
In a conductor, electric current can flow freely, in an insulator it cannot. Metals such as copper typify conductors, while most non-metallic solids are said to be good insulators, having extremely high resistance to the flow of charge through them. "Conductor" implies that the outer electrons of the atoms are loosely bound and free to move through the material. Most atoms hold on to their electrons tightly and are insulators. In copper, the valence electrons are essentially free and strongly repel each other. Any external influence which moves one of them will cause a repulsion of other electrons which propagates, "domino fashion" through the conductor
Force in experienced by an electron in a E field
F e E
Drift velocity
vd e E emobility of an electron
J e e E efree electron charge density (negative value)
J E conductivity (sigma) measured in siemens / m
Metallic Conductors
Assume that J and E are uniform
I SJ del
d J S
Vaba
b
LE del
d Ea
b
L1
d E Lba E Lab
or V E L
JI
S E
V
L
VL
SI V I R
RL
S RVab
I
a
b
LE del
d
S E del
d
Conductor Properties and Boundary Conditions
Boundary ConditionsConductor-free space boundary in electrostatic
1 – The static electric field intensity inside a conductor is zero2 – The static electric field at the surface of a conductor is everywhere directed normal to that surface3 – The conductor surface is an equipotential surface
E=0 within the conductor
LEdel
dEt w ENatb
1
2 h
ENatah 0 h 0 w finite
Et w 0 Et 0
SS del
d Q
Stop
d Sbottom
d Ssides
d Q
DN S Q S S DN S
Dt Et 0
DN 0 EN S
The Method of Images
The Method of Images
The Method of Images
Semiconductors
e e h h
Conductivity is a function of both hole and electron concentrations and mobility
Semiconductor Materials
Semiconductor Bandgap Energy EG (eV)Carbon (Diamond) 5.47Silicon 1.12 Germanium 0.66Tin 0.082Gallium Arsenide 1.42Indium Phosphide 1.35Boron Nitride 7.50Silicon Carbide 3.00Cadmium Selenide 1.70
Conductors, Dielectrics, Semiconductors
IIIA IVA VA VIA
10.8115
BBoron
12.011156
CCarbon
14.00677
NNitrogen
15.99948
OOxygen
IIB
26.981513
AlAluminum
28.08614
SiSilicon
30.973815
PPhosphorus
32.06416
SSulfur
65.3730
ZnZinc
69.7231
GaGallium
72.5932
GeGermanium
74.92233
AsArsenic
78.9634
SeSelenium
112.4048
CdCadmium
114.8249
InIndium
118.6950
SnTin
121.7551
SbAntimony
127.6052
TeTellurium
200.5980
HgMercury
204.3781
TiThallium
207.1982
PbLead
208.98083
BiBismuth
(210)84
PoPolonium
Portion of the Periodic Table Including the Most Important Semiconductor Elements
Conductors, Dielectrics, Semiconductors
Siwithanelectron
Si Si
SiSiSi
Si Si Si
Covalent bond filled
Two-dimensional silicon lattice with shared covalent bonds. At temperatures approaching 0 K, all bonds are filled, and the outer shells of the silicon atoms are completely full.
Conductors, Dielectrics, Semiconductors
Si Si Si
SiSiSi
Si Si Si
Free Electron
Hole (+q)
(-q)
An electron-hole pair is generated whenever a covalent bond is broken
Conductors, Dielectrics, Semiconductors
Si Si Si
SiPSi
Si Si Si
- q+ q
An extra electron is available from a phosphorus donor
atom
Conductors, Dielectrics, Semiconductors
Si Si Si
SiBSi
Si Si Si
Vacancy
Covalent bond vacancy from boron acceptor atom
Conductors, Dielectrics, Semiconductors
A silicon crystal is somewhat different from an insulator because at any temperature above absolute zero temperature, there is a finite probability that an electron in the lattice will be knocked loose from its position, leaving behind an electron deficiency called a "hole".
If a voltage is applied, then both the electron and the hole can contribute to a small current flow. The term intrinsic here distinguishes between the properties of pure "intrinsic" silicon and the dramatically different properties of doped n-type or p-type semiconductors.
Semiconductors
The Nature of Dielectric Materials
Most solid materials are classified as insulators because they offer very large resistance to the flow of electric current. Metals are classified as conductors because their outer electrons are not tightly bound, but in most materials even the outermost electrons are so tightly bound that there is essentially zero electron flow through them with ordinary voltages. Some materials are particularly good insulators and can be characterized by their high resistivities:
Resistivity (ohm)Glass 10^12
Mica 9 x 10^13
Quartz (fused) 5 x 10^16
Resistivity (ohm)Copper 1.7 x 10^-8
-Dielectric in an electric field can be viewed as a free-space arrangement of microscopic electric dipoles which are composed of positive and negative charges whose centers do not quite coincide.
Not free charges - They bound chargesThey are sources of electrostatic fields
Model – Polarization P and Permittivity
If a material contains polar molecules, they will generally be in random orientations when no electric field is applied. An applied electric field will polarize the material by orienting the dipole moments of polar molecules. This decreases the effective electric field between the plates and will increase the capacitance of the parallel plate structure. The dielectric must be a good electric insulator so as to minimize any DC leakage current through a capacitor.
Polar molecules have a permanent displacement existing between the centers of gravity of the positive and negative charges, and each pair of charges acts as a dipole. Dipoles are oriented randomly.
A non-polar molecule does not have this dipole arrangement until a field is applied.
A dipole may be described by its dipole moment p
p = Qd where d is the vector from the negative to the positive charge. p are in coulomb-metersP = Polarization
The Nature of Dielectric Materials
p total1
nv
i
p i
P0v
1
v1
nv
i
p i
lim
Electrical Properties
ASTM Standard
Unit Teflon®
PTFEPowderPaste
Disper.
Teflon® FEP
Teflon® PFA
Tefzel®
Dielectric Constant D150 1 MHz 2.1 2.1 2.1 2.6
Dissipation Factor D150 1MHz <0.0001 0.0006 0.0001 0.007
Arc Resistance D495 sec >300 >300 >180 122
Volume Resistivity D257 ohm·cm >1018 >1018 >1018 >1017
Surface Resistivity D257 ohm·sq >1018 >1016 >1017 >1015
The Nature of Dielectric Materials
Electrical Properties of Kapton®Type HN Polyimide Film
The Nature of Dielectric Materials
Property
Property Value--Film Thickness, mil (µm)
0.30 (7.6)
0.50 (12.7)*
1.00 (25.4)*
2.00 (50.8)*
3.00 (76.2)*
5.00 (127)*
Dielectric Strength, AC V/mil (kV/mm), min.
3,000(118)
3,000(118)
6,000(236)
5,000(197)
4,500(177)
3,000(118)
Volume Resistivity, ohm-cm at 200°C (392°F), min.
1012 1012 1012 1012 1012 1012
Dielectric Constant at 1 kHz, max.
4.0 4.0 3.9 3.9 3.9 3.9
Dissipation Factor at 1 kHz, max.
0.0070 0.0050 0.0036 0.0036 0.0036 0.0036
The Nature of Dielectric Materials
net total charge which crosses the elemental surface in an upward direction
Qb nQd S
Qb P S
Qb SPdot
d
Total enclosed charges Q = free charges
QT S0 Edot
d QT Qb Q
Q QT Qb S0 E P dot
d
D 0 E P
Boundary Conditions For Perfect Dielectric Materials
LE del
d
Etan1w Etan2w 0
Etan1 Etan2
If the tangential electric field intensity is continousacross the boundary, then tangential D is discontinuous
Dtan1
1Etan1
Dtan2
2
Dtan1
Dtan2
1
2
DN1S DN2S Q SS
DN1 DN2 S
In perfect dielectricsS 0
DN1 DN2
which follows that1EN1 2EN2
Normal of D is continous, normal of E is discontinous
Boundary Conditions For Perfect Dielectric Materials
The Nature of Dielectric Materials
the capacitance is independent of thepotential and the total charge, for their ratio constant
C = measured in farads (F) - one coulomb per volt
CQ
V0
C
SEdot
d
neg
pos
LEdot
d
Capacitance
Capacitance of two conductor systems as the ratio of the magnitude of the total charge on either conductor to the ratio of the potential difference between conductors
Capacitance
V0upper
lower
LEdot
d
d
0
zS
dS
d
Q S S
CQ
V0
Sd
Several Capacitance Examples
Coaxial Cable
Two concentric spheres
Parallel-plate capacitor – two dielectrics
V ab
L
2 0l n
b
a
pot ent ial dif f er ence bet ween point s= a and = b ( equat ion 11 sec. 4. 3)
C2 L
l nb
a
V abQ
4
1
a
1
b
CQ
V ab
4
1
a
1
b
C1
1
C 1
1
C 2
Capacitance of A Two-Wire Line
VL
2 ln
R0
R
VL
2 ln
R10
R1
lnR20
R2
L
2 ln
R10 R2
R20 R2
CL L
V
2 L
ln hh
2b
2b
Coil Modeling - Parameter Computation
d
d
AC r 0
12),
241(
22
2
2 cb
R
cRr
iii
Simplified Frequency Model of Coil
Capacitance
Capacitance – Zero-Potential Conducting Plane and Conducting Cylinder