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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

Engineering 45. Electrical Properties-1. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. Learning Goals – Elect. Props. How Are Electrical Conductance And Resistance Characterized?

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

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  1. Engineering 45 ElectricalProperties-1 Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. Learning Goals – Elect. Props • How Are Electrical Conductance And Resistance Characterized? • What Are The Physical Phenomena That Distinguish Conductors, SemiConductors, and NonConductors (i.e., Insulators)? • For Metals, How Is Conductivity Affected By Imperfections, Temp, and Deformation? • For Semiconductors, How Is Conductivity Affected By Impurities (Doping) And Temp?

  3. Georg Simon Ohm (1789-1854) First Stated a Relation for Electrical Current (I), and Electrical Potential (V) in Many Bulk Materials () Volt Meter Bulk Matl AmpMeter I Battery Electrical Conduction • The Constant of Proportionality, R, is • Called the Electrical RESISTANCE • Has units of Volts/Amps, a.k.a, Ohms (Ω) • This Expression is known as Ohm’s Law

  4. Fluid↔Current Flow Analogs Electrical Conduction cont. • Current as the “Electrical Fluid” • Wire as the “ Electrical Pipe” • Just as a Small Pipe “Resists” Fluid Flow, A Small Wire “Resists” current Flow • Thus Resistance is a Function of GEOMETRY and MATERIAL PROPERTIES • Next Discern the Resistance PROPERTY • Think of • Voltage as the “Electrical Pressure”

  5. Consider a Section of Physical Material, and Measure its Resistance Geometry Length X-Section Area Resistance, R Matl Prop → “” Area, A Length, L Electrical Resistivity • R↑ as L↑ • R  L • R↑ as A↓ • R  1/A • Thinking Physically, Since R is the Resistance to Current Flow, expect

  6. Thus Expect Resistance, R Matl Prop → “” Area, A Length, L Electrical Resistivity cont. • This is, in fact, found to be true for many Bulk Materials • Convert the Proportionality () to an Equality with the Proportionality Constant, ρ • Units for ρ • ρ → [Ω-m2]/m • ρ → Ω-m

  7. conductANCE is the inverse of resistANCE Conductance, G Matl Prop → “σ” Area, A Length, L Electrical Conductivity • Similarly, conductIVITY is the inverse of resistIVITY • Units for σ • σ = 1/ρ → 1/ Ω-m • Now Ω−1 is Called a Siemens, S • σ → S/m

  8. Recall Ohm’s Law L V Ohm Related Issues • J  Current Density in A/m2 • E  Electric Field in V/m • In the General Case • E = ρJ is the NORMALIZED, Resistive, Version of Ohm’s Law

  9. Recall Ohm’s Law L V Normalized, Conductive Ohm • ReArranging • G is Conductance • Recall also • J = σE is the Normalized, Conductive Version of Ohm’s Law

  10. Metals Conductivity (107 S/m) Some Conductivities in S/m • Metals  107 • SemiConductors • Si (intrinsic)  10−4 • Ge  100 = 1 • GaAs  10− 6 • InSb  104 • Insulators • SodaLime Glass  10− 11 • Alumina  10− 13 • Nylon  10−13 • Polyethylene  10−16 • PTFE  10−17

  11. Conductivity Example • What is the minimum diameter (D) of a 100m wire so that ΔV < 1.5 V while carrying 2.5A? 100m - e I = 2.5A + - Cu wire DV • Recall • Also G by σ & Geometry • For Cu: σ = 6.07x107 S/m

  12. Conductivity Example cont • What is the minimum diameter (D) of a 100m wire so that ΔV < 1.5 V while carrying 2.5A? 100m - e I = 2.5A + - Cu wire DV • Solve for D • Sub for Values

  13. As noted In Chp2 Electrons in a FREE atom Can Reside in Quantized Energy Levels The Energy Levels Tend to be Widely Separated, Requiring significant Outside Energy To move an Electron to the next higher level Electronic Conduction • In The SOLID STATE, Nearby Atoms Distort the Energy LEVELS into Energy BANDS • Each Band Contains MANY, CLOSELY Spaced Levels

  14. Consider the 3s Energy Level, or Shell, of an Atom in the SOLID STATE with EQUILIBRIUM SPACING r0 Solid State Energy Band Theory By the Pauli Exclusion Principle Only ONE e− Can occupy a Given Energy Level

  15. The N atoms per m3 with Spacing r0 produces an Allowed-Energy BAND of Width ΔE Most Solids have N = 1028-1029 at/m3 Thus the ΔE wide Band Splits into 1029/m3 Allowed E-Levels Band Theory, cont. • Leads to a band of energies for each initial atomic energy level • e.g., 1s energy band for 1s energy level

  16. Given ΔE 15 eV N  5 x 1028 at/cu-m Then the difference between allowed Energy Levels within the Band, δE Energy Band Calc • The Thermal Energy at Rm Temp is 25 meV/at, or about 1026 times δE • Thus if bands are Not Completely Filled, e− can move easily between allowed levels

  17. In Metals The Electronic Energy Bands Take One of Two Configurations Partially Filled Bands e−can Easily move Up to Adjacent Levels, Which Frees Them from the Atomic Core Overlapping Bands e− can Easily move into the Adjacent Band, Which also Frees Them from the Atomic Core Energy Energy empty band empty GAP band partly filled filled valence valence band band filled states filled states filled filled band band Electronic Conduction  Metals

  18. Atoms at Their Lowest Energy Condition are in the “Ground State”, and are Not Free to Leave the Atom Core In Metals, the Energy Supplied by Rm Temp Can move the e− to a Higher Level, making them Available for Conduction V+ E-Field V- Net e- Flow Current Flow Metal Conduction, Cont. • Metallic Conduction Model → Electron-Gas or Electron-Sea • Note: e−’s Flowing “UPhill” constitutes Current Flowing DOWNhill

  19. Energy Energy empty ConductionBand ConductionBand empty band band ? GAP GAP filled filled Valence valence band band filled states filled states filled filled band band 7 Insulators & Semiconductors • Insulators: • Higher energy states not accessible due to lg gap • Eg > ~3.5 eV • Semiconductors: • Higher energy states separated by smaller gap • Eg < ~3.5 eV

  20. The Two Basic Components of Solid-St Electronic Conduction The Number of FREE Electrons, n The Ease with Which the Free e−’s move Thru the Solid i.e. the electron Mobility, µe 6 Cu + 3.32 at%Ni 5 -m) Cu + 2.16 at%Ni 4 Resistivity, ρ deformed Cu + 1.12 at%Ni 3 -8 Cu + 1.12 at%Ni (10 2 “Pure” Cu 1 0 -200 -100 0 T (°C) Metals: ρvs T, ρvs Impurities • Consider The ρ Characteristics for Cu Metal & Alloys

  21. Since “Double Ionization” of Atom Cores is difficult n(Hi-T)  n(Lo-T) Thus T, Impurities and Defects must Cause Reduced µe These are all e- Scattering Sites Vacancies Grain Boundaries 6 Cu + 3.32 at%Ni 5 -m) Cu + 2.16 at%Ni 4 Resistivity, ρ deformed Cu + 1.12 at%Ni 3 -8 Cu + 1.12 at%Ni (10 2 “Pure” Cu 1 0 -200 -100 0 T (°C) Metals -  vs T, Impurities cont • Impurities; e.g., Ni above • Dislocations; e.g., deformed

  22. The Data Shows The Factors that Reduce σ Higher Temperature Impurities Defects These Affects are PARALLEL Processes i.e., They Act Largely independently of each other Metal  - Mathiessen’s Rule • The Cumulative Effect of ||-Processes is Calculated by Mathiessen’s Rule of Reciprocal Addition

  23. Temperature Affects may be approximated with a Linear Expression Resistivity Relations for Metals • For A Single Impurity That Forms a Solid-Solution • Where • A is an Alloy-Specific Constant, Ω-m/at-frac • ci is the impurity Concentration in in the atomic-fraction Format • At-frac = at%x(1/100%) • Where • 0 is the Resisitivity at the Baseline Temperature, Ω-m • a is the Slope of ρvs T Curve, Ω-m/K

  24. In alloys where the impurity results, not in Solid-Solution, but in the Formation of a 2nd Xtal Structure, or Phase, Use a Rule-of-Mixtures Relation for ρi Use Vol-Fractions as the Weighting Factor ρRelations for Metals cont • Where • ρk is the Resistivity of phase-k • Vk is the Volume-Fraction of phase-k • Plastic Deformation • There is no Simple Relation for This • Consult individual metal or alloy data

  25. Est. σfor a Cu-Ni alloy with yield strength of 125 MPa From Fig 7.16 Find Composition for Sy = 125 MPa 180 160 140 strength (MPa) 120 10 0 21 wt%Ni 8 0 Yield 60 0 10 2 0 3 0 4 0 5 0 wt. %Ni, (Concentration C) r 5 0 Ohm-m) 40 3 0 Resistivity, -8 2 0 (10 1 0 0 0 10 2 0 3 0 4 0 5 0 wt. %Ni, (Concentration C) Example  Estimate σ • So need 21 wt% Ni • Find ρfrom Fig 18.9 • ρ 30x10-8Ω-m • And σ= 1/ρ •  σ= 3.3x106 S/m

  26. All Done for Today UsingBandGaps To Make LEDs

  27. Chabot Engineering Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  28. http://www.chemistry.wustl.edu/~edudev/LabTutorials/PeriodicProperties/MetalBonding/MetalBonding.htmlhttp://www.chemistry.wustl.edu/~edudev/LabTutorials/PeriodicProperties/MetalBonding/MetalBonding.html

  29. WhiteBoard Work • Derive Relation for e- Drift Velocity, vd • Calculate the Drift Velocity in a 20 foot Gold Wire Connected to a 9Vdc Batt • Assume Au Atoms in the Solid Are Singly Ionized, contributing 1 conduction-e- per atom (monovalent) • Compare (random) THERMAL Velocity

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