Advanced Electrical Engineering
Advanced Electrical Engineering
Michael E. Auer
Electrostatics
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
AEE Content
Advanced Circuit Analysis
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Basic Concepts
Three-Phase Circuits
Transforms
Power Conversion and Management
Field Theory
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Waves and Vector Fields
Transmission Line Theory
• Electrostatics
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Magnetostatics
Applications
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Michael E.Auer
Magnetic Field Applications
Basics of Electrical Machines
18.06.2012
AEE07
Advanced Electrical Engineering
Content
Michael E.Auer
•
Maxwell’s Equations
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Coulomb’s Law
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Gauss’s Law
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Electrical Potential
•
Resistance of Conductors
•
Dielectrics and Capacitance
18.06.2012
AEE07
Advanced Electrical Engineering
Content
Michael E.Auer
•
Maxwell’s Equations
•
Coulomb’s Law
•
Gauss’s Law
•
Electrical Potential
•
Resistance of Conductors
•
Dielectrics and Capacitance
18.06.2012
AEE07
Advanced Electrical Engineering
Maxwell’s Equations in Differential Form
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Maxwell’s Equations in Integral Form
∫ H ⋅ ds = ∫ J ⋅ dA
∂B
∫ E ⋅ ds = − ∫ ∂ t ⋅ dA
Ampere's law
Law of Induction
(H swirling around S)
(E swirling around – dB/dt)
∫ B ⋅ dA = 0
∫ D ⋅ dA = ∫ ρ ⋅ dV
(B is source-free)
Material Equations
Furthermore
(Charges are sources of D)
∂D
J = ∫ u ⋅ dρ +
∂t
convection current
Michael E.Auer
D =ε ⋅E
B=µ ⋅H
18.06.2012
displacement current
AEE07
Advanced Electrical Engineering
Permitivity and Magnetic Permability
ε = ε0 ⋅εr
Velocity
Michael E.Auer
µ = µ0 ⋅ µ r
1
u=
ε ⋅µ
or
c
u=
ε r ⋅ µr
18.06.2012
with
c=
1
ε 0 ⋅ µ0
AEE07
Advanced Electrical Engineering
Current Density
For a surface with any orientation:
J is called the current density
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Content
Michael E.Auer
•
Maxwell’s Equations
•
Coulomb’s Law
•
Gauss’s Law
•
Electrical Potential
•
Resistance of Conductors
•
Dielectrics and Capacitance
18.06.2012
AEE07
Advanced Electrical Engineering
Coulomb‘s Law
Electric field at point P due to single charge
Electric force on a test charge placed at P
Electric flux density D
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Electric Field due to two Charges
Multiple charges
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Example
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Possible Charge Distributions
Field due to:
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Content
Michael E.Auer
•
Maxwell’s Equations
•
Coulomb’s Law
•
Gauss’s Law
•
Electrical Potential
•
Resistance of Conductors
•
Dielectrics and Capacitance
18.06.2012
AEE07
Advanced Electrical Engineering
Gauss‘s Law
Application of the divergence theorem
gives:
And therefore
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Application of Gauss‘s Law
Electric field of an
infinite line charge
Michael E.Auer
Construct an imaginary Gaussian
cylinder of radius r and height h:
18.06.2012
AEE07
Advanced Electrical Engineering
Content
Michael E.Auer
•
Maxwell’s Equations
•
Coulomb’s Law
•
Gauss’s Law
•
Electrical Potential
•
Resistance of Conductors
•
Dielectrics and Capacitance
18.06.2012
AEE07
Advanced Electrical Engineering
Electric Potential (1)
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Electric Potential (2)
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Electric Potential due to Charges
For a point charge, V at range R is:
In electric circuits, we usually select a
convenient node that we call ground and
assign it zero reference voltage. In free
space and material media, we choose
infinity as reference with V = 0. Hence,
at a point P
Michael E.Auer
18.06.2012
For continuous charge distributions:
AEE07
Advanced Electrical Engineering
Relating E to V
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Content
Michael E.Auer
•
Maxwell’s Equations
•
Coulomb’s Law
•
Gauss’s Law
•
Electrical Potential
•
Resistance of Conductors
•
Dielectrics and Capacitance
18.06.2012
AEE07
Advanced Electrical Engineering
Conduction Current
Conduction current density:
Generalized Ohm’s Law
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Resistance
Longitudinal Resistor
For any conductor:
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Conductance of a Coaxial Cable
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Content
Michael E.Auer
•
Maxwell’s Equations
•
Coulomb’s Law
•
Gauss’s Law
•
Electrical Potential
•
Resistance of Conductors
•
Dielectrics and Capacitance
18.06.2012
AEE07
Advanced Electrical Engineering
Electric Breakdown
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Boundary Conditions (1)
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Boundary Conditions (2)
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Boundary Conditions Example
At conductor boundary,
E field direction is always
perpendicular to conductor surface
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Capacitance
For any two-conductor configuration:
For any resistor:
Michael E.Auer
18.06.2012
AEE07
Advanced Electrical Engineering
Examples
Parallel Plate Capacitor
Coaxial Capacitor
Michael E.Auer
18.06.2012
AEE07