EEE 498/598 Overview of Electrical Engineering

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EEE 498/598 Overview of Electrical Engineering. Lecture 9: Faraday’s Law Of Electromagnetic Induction; Displacement Current; Complex Permittivity and Permeability. Lecture 9 Objectives.

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### EEE 498/598Overview of Electrical Engineering

Lecture 9: Faraday’s Law Of Electromagnetic Induction; Displacement Current; Complex Permittivity and Permeability

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Lecture 9 Objectives
• To study Faraday’s law of electromagnetic induction; displacement current; and complex permittivity and permeability.

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• In the static case (no time variation), the electric field (specified by E and D) and the magnetic field (specified by B and H) are described by separate and independent sets of equations.
• In a conducting medium, both electrostatic and magnetostatic fields can exist, and are coupled through the Ohm’s law (J = sE). Such a situation is called electromagnetostatic.

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Electromagnetostatic Fields
• In an electromagnetostatic field, the electric field is completely determined by the stationary charges present in the system, and the magnetic field is completely determined by the current.
• The magnetic field does not enter into the calculation of the electric field, nor does the electric field enter into the calculation of the magnetic field.

6

The Three Experimental Pillars of Electromagnetics
• Electric charges attract/repel each other as described by Coulomb’s law.
• Current-carrying wires attract/repel each other as described by Ampere’s law of force.
• Magnetic fields that change with time induce electromotive force as described by Faraday’s law.

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toroidal iron

core

switch

compass

battery

secondary

coil

primary

coil

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• Upon closing the switch, current begins to flow in the primary coil.
• A momentary deflection of the compassneedle indicates a brief surge of current flowing in the secondary coil.
• The compass needle quickly settles back to zero.
• Upon opening the switch, another brief deflection of the compass needle is observed.

9

S

C

• “The electromotive force induced around a closed loop C is equal to the time rate of decrease of the magnetic flux linking the loop.”

10

Faraday’s Law of Electromagnetic Induction (Cont’d)
• S is any surface bounded by C

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Stokes’s theorem

assuming a stationary surface S

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• Since the above must hold for any S, we have

differential form of Faraday’s law (assuming a stationary frame of reference)

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• Faraday’s law states that a changing magnetic field induces an electric field.
• The induced electric field is non-conservative.

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Lenz’s Law
• “The sense of the emf induced by the time-varying magnetic flux is such that any current it produces tends to set up a magnetic field that opposes the change in the original magnetic field.”
• Lenz’s law is a consequence of conservation of energy.

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• “The electromotive force induced around a closed loop C is equal to the time rate of decrease of the magnetic flux linking the loop.”
• For a coil of N tightly wound turns

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S

C

• S is any surface bounded by C

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• Faraday’s law applies to situations where
• (1) the B-field is a function of time
• (2) ds is a function of time
• (3) B and ds are functions of time

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• The induced emf around a circuit can be separated into two terms:
• (1) due to the time-rate of change of the B-field (transformer emf)
• (2) due to the motion of the circuit (motional emf)

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transformer emf

motional emf

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2

-

B

v

+

1

Moving Conductor in a Static Magnetic Field
• Consider a conducting bar moving with velocity v in a magnetostatic field:
• The magnetic force on an electron in the conducting bar is given by

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2

-

B

v

+

1

Moving Conductor in a Static Magnetic Field (Cont’d)
• Electrons are pulled toward end 2. End 2 becomes negatively charged and end 1 becomes + charged.
• An electrostatic force of attraction is established between the two ends of the bar.

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Moving Conductor in a Static Magnetic Field (Cont’d)
• The electrostatic force on an electron due to the induced electrostatic field is given by
• The migration of electrons stops (equilibrium is established) when

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Moving Conductor in a Static Magnetic Field (Cont’d)
• A motional(or “flux cutting”) emfis produced given by

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Electric Field in Terms of Potential Functions
• Electrostatics:

scalar electric potential

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Electric Field in Terms of Potential Functions (Cont’d)
• Electrodynamics:

vector magnetic potential

• both of these potentials are now functions of time.

scalar electric potential

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Ampere’s Law and the Continuity Equation
• The differential form of Ampere’s law in the static case is
• The continuity equation is

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Ampere’s Law and the Continuity Equation (Cont’d)
• In the time-varying case, Ampere’s law in the above form is inconsistent with the continuity equation

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Ampere’s Law and the Continuity Equation (Cont’d)
• To resolve this inconsistency, Maxwell modified Ampere’s law to read

displacement current density

conduction current density

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Ampere’s Law and the Continuity Equation (Cont’d)
• The new form of Ampere’s law is consistent with the continuity equation as well as with the differential form of Gauss’s law

qev

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Displacement Current
• Ampere’s law can be written as

where

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Displacement Current (Cont’d)
• Displacement current is the type of current that flows between the plates of a capacitor.
• Displacement current is the mechanism which allows electromagnetic waves to propagate in a non-conducting medium.
• Displacement current is a consequence of the three experimental pillars of electromagnetics.

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z

A

ic

+

z = d

e

id

z = 0

-

Displacement Current in a Capacitor
• Consider a parallel-plate capacitor with plates of area A separated by a dielectric of permittivity e and thickness d and connected to an ac generator:

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Displacement Current in a Capacitor (Cont’d)
• The electric field and displacement flux density in the capacitor is given by
• The displacement current density is given by
• assume fringing is negligible

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Displacement Current in a Capacitor (Cont’d)
• The displacement current is given by

conduction current in wire

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Conduction to Displacement Current Ratio
• Consider a conducting medium characterized by conductivity s and permittivity e.
• The conduction current density is given by
• The displacement current density is given by

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Conduction to Displacement Current Ratio (Cont’d)
• Assume that the electric field is a sinusoidal function of time:
• Then,

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Conduction to Displacement Current Ratio (Cont’d)
• The value of the quantity s/we at a specified frequency determines the properties of the medium at that given frequency.
• In a metallic conductor, the displacement current is negligible below optical frequencies.
• In free space (or other perfect dielectric), the conduction current is zero and only displacement current can exist.

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Complex Permittivity
• In a good insulator, the conduction current (due to non-zero s) is usually negligible.
• However, at high frequencies, the rapidly varying electric field has to do work against molecular forces in alternately polarizing the bound electrons.
• The result is that Pis not necessarily in phase with E, and the electric susceptibility, and hence the dielectric constant, are complex.

42

Complex Permittivity (Cont’d)
• The complex dielectric constant can be written as
• Substituting the complex dielectric constant into the differential frequency-domain form of Ampere’s law, we have

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Complex Permittivity (Cont’d)
• Thus, the imaginary part of the complex permittivity leads to a volume current density term that is in phase with the electric field, as if the material had an effective conductivity given by
• The power dissipated per unit volume in the medium is given by

44

Complex Permittivity (Cont’d)
• The term we E2 is the basis for microwave heating of dielectric materials.
• Often in dielectric materials, we do not distinguish between s and we, and lump them together in we as
• The value of seff is often determined by measurements.

45

Complex Permittivity (Cont’d)
• In general, both e and e depend on frequency, exhibiting resonance characteristics at several frequencies.

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Complex Permittivity (Cont’d)
• In tabulating the dielectric properties of materials, it is customary to specify the real part of the dielectric constant (e / e0) and the loss tangent (tand) defined as

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Complex Permeability
• Like the electric field, the magnetic field encounters molecular forces which require work to overcome in magnetizing the material.
• In analogy with permittivity, the permeability can also be complex

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Maxwell’s Curl Equations for Time-Harmonic Fields in Simple Medium Using Complex Permittivity and Permeability

complex

permeability

complex

permittivity

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