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1. WAVES & PHASORS EE341 – Aaron Scher – Oregon Tech

2-D Array of a Liquid Crystal Display

Examples of EM Applications

Dimensions and Units

Fundamental Forces of Nature

Gravitational Force

Force exerted on mass 2 by mass 1

Gravitational field induced by mass 1

Charge: Electrical property of particles

1 coulomb represents the charge on ~ 6.241 x 1018 electrons

Charge of an electron

Units: coulomb

e = - 1.602 x 10-19 C

Charge conservation

Cannot create or destroy charge, only transfer

One coulomb: amount of charge accumulated in one second by a current of one ampere.

The coulomb is named for a French physicist, Charles-Augustin de Coulomb (1736-1806), who was the first to measure accurately the forces exerted between electric charges.

Electrical Force

Force exerted on charge 2 by charge 1

Electric Field In Free Space

Permittivity of free space

Electric Field Inside Dielectric Medium

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:

Magnetic Field

Magnetic field induced by a current in a long wire

Magnetic permeability of free space

Electric and magnetic fields are connected through the speed of light:

Electric charges can be isolated, but magnetic poles always exist in pairs.

Static vs. Dynamic

Static conditions: charges are stationary or moving, but if moving, they do so at a constant velocity.

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

Material Properties

Traveling Waves

¨  Waves carry energy ¨  Waves have velocity ¨  Many waves are linear: they do not affect the

passage of other waves; they can pass right through them

¨  Transient waves: caused by sudden disturbance ¨  Continuous periodic waves: repetitive source

Types of Waves

Sinusoidal Waves in Lossless Media

y = height of water surface x = distance

http://phet.colorado.edu/sims/html/wave-on-a- string/latest/wave-on-a-string_en.html

Traveling wave animation:

Phase velocity

If we select a fixed height y0 and follow its progress, then

=

Wave Frequency and Period

Direction of Wave Travel

Wave travelling in +x direction

Wave travelling in ‒x direction

+x direction: if coefficients of t and x have opposite signs

‒x direction: if coefficients of t and x have same sign (both positive or both negative)

Phase Lead & Lag

Wave Travel in Lossy Media

Attenuation factor

Example 1-1: Sound Wave in Water

Given: sinusoidal sound wave traveling in the positive x-direction in water Wave amplitude is 10 N/m2, and p(x, t) was observed to be at its maximum value at t = 0 and x = 0.25 m. Also f=1 kHz, up=1.5 km/s. Determine: p(x,t) Solution:

The EM Spectrum

Complex Numbers

We will find it is useful to represent sinusoids as complex numbers

jyxz += θθ jezzz =∠=

1−=j

Rectangular coordinates

Polar coordinates

θθθ sincos je j ±=±

Relations based on Euler’s Identity

( ) yz xz

=

=

)Im( Re

Relations for Complex Numbers

Learn how to perform these with your calculator/ computer

Phasor Domain

Transformations are used in engineering to simplify physical calculations

Phasors are complex numbers that are used to characterize the amplitudes and phases of time-harmonic physical quantities.

A sinusoidal input to an LTI system will result in a sinusoidal output at the same frequency.

The effects of any LTI system can be represented with phasor multiplication.

Phasor transformations may be used on vector functions, multivariable functions, and functions of 3D space.

Phasor Domain

Phasor counterpart of

Time and Phasor Domain

It is much easier to deal with exponentials in the phasor domain than sinusoidal relations in the time domain Just need to track magnitude/phase, knowing that everything is at frequency ω