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ECE 107: ElectromagnetismNotes Set 1

Instructor: Prof. Vitaliy LomakinDepartment of Electrical and Computer Engineering

University of California, San Diego, CA 92093

Introduction (1)

1 212 2

0 12

1ˆ4e

q qRπε

=F R

atomElectromagnetism (1 or 4 interactions)Nuclear (strong) interactionWeak interaction

Nuclear and particle physics

Gravity Electromagnetic

G gravitational constant ε0 permittivity in free space

Coulomb’s Law

Introduction (2)

For an electron: Fe /Fg = 4x1042, i.e. the electric force is much stronger!

always attractive attractive/repulsivesmall largeno control extensive control

Gravity Electromagnetic

Introduction (3)

Why does gravity matter?

•Conservation of charge: Charges can’t be created or destroyed•On average the universe is neutral

Electric fields• Electric field due to a charge:

2( )C mε=D E

- ε = ε r ε0 - permittivity- Characterizes the effect due to a charge - Can exists in space where no charges are present (still excited by a charge)ε0 = 8.854x10-12 F/m

• Electric flux density:

Electric fields (2)linear superposition

electric dipole

Magnetic fields (1)

• Magnet has poles, which always exist in pairs (unlike electric charges, which can be isolated)

• Magnetic field lines have neither a start nor an end

• Magnetic field also can be created by electric currents

• Magnetic interactions between metallic materials were known for centuries (although quantum mechanical in nature)

• If a needle is placed near a magnet, it aligns along lines called magnetic-field lines

Magnetic fields (2)• Magnetic flux density induced by a

steady current in the z direction

• - permeability

• Magnetic field: such that

• Amazingly - speed of light

( ),T Tesla

( )A mH

Magnetic permeability of free space

µ=B H

Static and dynamic fields (1)• Electric field is excited by • a charge . • Magnetic field is excited by • a current . • For a constant current I the electric and magnetic fields are

independent from each other. • Electrostatics and magnetostatics correspond to stationary charges

and currents, respectively. Electrostatic and magnetostatic fields are uncoupled

• Dynamics corresponds to time varying fields. Dynamic magnetic and electric fields depend on each other. A dynamic electric field generates a dynamic magnetic field and vice versa.

q

I dq dt=

Static and dynamic fields (2)•Time varying electric field

Easiest way is to move a charge

This leads to a time varying current and a time varying magnetic field.

Static and dynamic fields (3)

Under static conditions, electric and magnetic fields are independent, but under dynamic conditions, they become coupled.

Maxwell’s Equations

(1831 – 1879)

0

t

tρ

∂∇× = −

∂∂

∇× = +∂

∇ ⋅ =∇ ⋅ =

BE

DH J

DB

“From a long view of the history of mankind — seen from, say, ten thousand years from now, there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics”. –R. P. Feynman

Material characterization• Properties of materials are characterized by permittivity,

permeability, and conductivity• electric permittivity• magnetic permeability• conductivity of a material

characterizes how easy charges can move in a material• perfect dielectric• perfect conductor

These parameters depend on frequency, temperature, direction, …

( 1 ( ))S m mσ = Ω

σ = ∞

εµ

Material characterization (2)Polarization of atoms changes electric field

New field can be accounted for by changing the permittivity

Permittivity of the material

Another quantity used in EM is the electric flux density D:

Similar arguments for permeability (μ) B = μH where μr >> 1 for magnetic materials

λ = c/frequency

Electromagnetics is a foundation of electrical engineering

Low Frequency

Circuit

High Frequency

Optics

Ultra-high Frequency

X/Gamma-rays

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Electromagnetics is new & exciting

• Wireless communication:o Radio wave propagationo Antenna design

• High-speed high-density circuits:o EM mutual couplingo EM compatibility

• Defense:o Stealth technologyo EM missiles

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Stealth technology

F117 Night Hawk B-2 Spirit

F-22 Raptor Joint Strike Fighter

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• Remote sensing:o Earth surface remote sensingo Atmospheric remote sensingo Nondestructive testingo Ground penetrating radar

• Medical application:o Magnetic resonance imagingo Microwave imagingo EM hyperthermia

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MRI System

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• Photonics:o Waveguides and fiberso Resonatorso Laserso Filterso Couplerso Etc, etc, etc

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Photonic waveguides and lasers

Magnetic storage and memory

UCT)

ATE

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Traveling waves (1)• Waves carry energy from one point in space to another• Examples of waves: Ocean waves, sound waves,

mechanical waves on strings, electromagnetic waves • Waves of completely different physical nature have many

common properties• Waves have finite velocity needed to travel from a point to

a point. For example, sound waves in air have velocity of 330 m/s, velocity of light is 3x108 m/s

• When the strength of the field associated with a wave depends on the source linearly, then two waves travel independently from each other

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Traveling waves (2)• Transient waves are caused by a short-duration source• Harmonic waves are generated by a continuously

oscillating source• Some possible traveling wave types:

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Traveling waves (3): Pulsed waves

• Consider a one-dimensional wave caused by a pulsed source

• Let y be the vertical wave displacement and x be the direction of the wave propagation. Then, neglecting any possible loss of energy carried by the wave,

where is the excitation functionis the amplitudeis the velocity

( , ) ( )y x t Af t x u= −

( )f tA

0( )f t x u−

0x t u=

1x t u=

2x t u=

xy

u

1( )f t x u−

2( )f t x u−

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Traveling waves (4): Sinusoidal waves

• As an example, assume a one-dimensional ocean wave• Let y be the vertical wave displacement and x be the

direction of the wave propagation. Then, neglecting any possible loss of energy carried by the wave,

• A – amplitude • T – period (s)• λ – wavelength (m)•• - reference phase

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Traveling waves (5): Sinusoidal waves

• Reference phase

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• Phase••

• - phase velocity • - frequency

•

• - angular frequency• - wavenumber

Traveling waves (6): Sinusoidal waves

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Traveling waves (7): Lossy medium

• When loss is present the wave is attenuated–– - the attenuation constant , ( ), neper per meterNp mα

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Review of phasors (1)• Consider a function:

• It can be shown that

• - phasor • Phasors contain the information about the amplitude AND

phase, while the time dependence is known as • Use of phasors simplify solutions for harmonic fields

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Review of phasors (2)

Review of phasors (3)

Time Domain Frequency Domain( )φωυ +== tRIiR cosm

φ∠= mRIV

Current through resistor

( )φω += tIi cosm

Time domain

Phasor Domain

Review of phasors (4)

Time Domain

Time domain

Phasor Domain

Review of phasors (5)

Time Domain

Time domain

Phasor Domain

Review of phasors (6)

Review of phasors (7)

Cont.

Review of phasors (8)

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Review of phasors (9)• Traveling waves in terms of phasors:

• +x traveling wave• -x traveling wave

Why 50 Ohms?

Transmission lines…