Download ppt - Chapter 5 Conductors, Dielectrics and Capacitance Current and Current Density

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


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



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


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


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.


Si Si Si

SiSiSi

Si Si Si

Free Electron

Hole (+q)

(-q)

An electron-hole pair is generated whenever a covalent bond is broken


Si Si Si

SiPSi

Si Si Si

- q+ q

An extra electron is available from a phosphorus donor

atom


Si Si Si

SiBSi

Si Si Si

Vacancy

Covalent bond vacancy from boron acceptor atom


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


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


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Electrical Properties of Kapton®Type HN Polyimide Film


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


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 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