ECE 1100: Introduction toECE 1100: Introduction toElectrical and Computer EngineeringElectrical and Computer Engineering

Sinusoidal SignalsWavest

v(t)

Wanda WosikAssociate Professor, ECE Dept.

Spring 2011

Slides developed by Dr. Jackson

Basic FactsBasic Facts

Sinusoidal waveforms (waves that vary sinusoidally in time) are the most important types of waveforms encountered in physics and engineering.

Most natural sources of radiation (the sun, etc.) emit sinusoidal waveforms.

Most human-made systems produce sinusoidal waveforms (AC generators, microwave oscillators, etc.)

Most communications is done via sinusoidal waveforms that have been modulated, either in an analog fashion (such as AM or FM) or digitally.

General Sinusoidal WaveformGeneral Sinusoidal Waveform

cosv t A t

A = amplitude of sinusoidal waveform = “radian frequency” of sinusoidal waveform [radians/s] = phase of sinusoidal signal [radians]

t

v (t)

A

-Amax /t

Period of Sinusoidal WavePeriod of Sinusoidal Wave

T = period (cycle) of wave [s] = time it takes for the waveform to repeat itself.

In this example, T = 0.5 [s].

t [s]

v (t)T

0.5 1.0 1.5

Frequency of Sinusoidal WaveFrequency of Sinusoidal Wave

f = frequency = # cycles (periods) / s Units: Hz = cycle/s

In this example, f = 2 Hz

f = 1/T [Hz]cycles/s = 1 / (s/cycle)

t [s]

v (t)

1.00.5

1 [s]

1.5

Radian FrequencyRadian Frequency

cosv t A t

Since the cosine function repeats after 2, we have

2T

t

v (t) T

= 2 f [rad/s]

Hence: or

2

T

Rotation with angular frequency

SummarySummary

cosv t A t

t

v(t)

A

-Amaxt /

= 2 f [rad/s]

f = 1/T [Hz]

WavesWaves

Waves in nature (and engineering) are usually sinusoidal in shape, and they move outward from the source with a velocity.

Waves (cont.)Waves (cont.)

We focus attention on a particular direction, called z.

h (z) = height of wave at a fixed time t = 0.

This is a “snapshot” of the wave at a fixed time t = 0.

z

cosh z A kz

z [m]

h (z)crest

trough

v = velocity

WavenumberWavenumber

Assumed form of wave: k is “wavenumber” of wave.

Wavelength Wavelength

z [m]

h(z)The wavelength is the distance it takes for the waveform to repeat (for a fixed time).

cosh z A kz 2k 2

k

Wave at Fixed Observation Point Wave at Fixed Observation Point

Next question: what is the velocity that the observer will feel?

z [m]

h (z)v = velocity

observer

z

€

h(z, t) = Acos(ωt − kz)

€

cos(ωΔt − kΔz) =1 Point on crest

€

t − kΔz = 0

€

z Δt = ω k

€

c = ωk =

ω

ω μ0ε 0µ0=4x10-7 H/m0=8.85x10-12 F/mc=2.997x108 m/sv = velocity

Wave at Fixed Observation Point Wave at Fixed Observation Point

Next question: what would the height as a function of time look

like, for an observer at a fixed value of z = z0?

z [m]

h (z)v = velocity

observer

We can pretend that the wave is fixed and the observer is

moving backwards at velocity v.

cosh z A kz

0z z vt

0cosh t A k z vt

Wave at Fixed Observation Point (cont.) Wave at Fixed Observation Point (cont.)

z [m]

h (z)v = velocity

observer

0

0

0

cos

cos

cos

h t A kz kv t

A kz kv t

A kv t kz

Wave at Fixed Observation Point (cont.) Wave at Fixed Observation Point (cont.)

z [m]

h (z)v = velocity

observer

0cosh t A k z vt

0cosh t A kv t kz

Define kv

0cosh t A t kz

Wave at Fixed Observation Point (cont.) Wave at Fixed Observation Point (cont.)

The observed amplitude varies sinusoidally in time!

General Form of Wave General Form of Wave

, cosh t z A t kz

z [m]

h (t, z) v = velocity

vk

2 f 1

fT

2

k

Allowing for both t and z to be arbitrary, we have.

General Form of Wave (cont.) General Form of Wave (cont.)

2

2 /

fv f

k

For the velocity we can also write

v fT

E

H

Electromagnetic WavesElectromagnetic Waves

Wave propagation with speed of light c

€

c =1

ε 0μ0

Electromagnetic WaveElectromagnetic Wave

There are two types of fields in nature: electric and magnetic.

An electromagnetic wave has both fields, perpendicular to each other, and it travels (propagates) at the speed of light.

You will learn much more about EM waves in ECE 3317.

velocity = c (speed of light)

H

E The power flows in the direction E H

Electromagnetic Wave (cont.)Electromagnetic Wave (cont.)

z

c = speed of lightelectric field vector

magnetic field vector

The amplitudes of the electric and magnetic fields vary sinusoidal in space (just like the amplitude of a water wave).

Electromagnetic Wave (cont.)Electromagnetic Wave (cont.)

zearth

x

power flow (z direction)

The electric field vector is in the direction of the transmitting antenna.

yH y H

xE x E

Transmitting AntennaTransmitting Antenna

AM Radio: 550 kHz < f < 1610 kHz

Ex

Electric field is vertically polarized

zearth

x

Transmitting Antenna (cont.)Transmitting Antenna (cont.)

FM Radio: 88 MHz < f < 108 MHz

Electric field is horizontally polarized

z

Ey

earth

x VHF TV: 55.25 MHz < f < 216 MHz

UHF TV: 470 MHz < f < 806 MHz

Equation for Electric FieldEquation for Electric Field

, cos [V/m]xE t z A t kz

z

c = speed of lightelectric field vectorx

Note: It is the electric field that would be received by a wire antenna.

Value of Value of kk

, cosxE t z A t kz

From Maxwell’s equations, it can be shown that:

0 0 [radians/m]k

70 4 10 Henrys / m

120 8.85418782 10 Farads / m

(permeability of free space)

(permittivity of free space)

Velocity of Wave (cont.)Velocity of Wave (cont.)

velocity = v = c = /k

Note: all frequencies travel the same speed.

From previous notes on waves (propagation in vacuum):

Hence we have

0 0

1c

c = 2.99792458 108 [m/s]

0 0

c

(exact defined quantity)

Summary of Wave FormulasSummary of Wave Formulas

k = 2 /

c = f

0 0

1c

c = 2.99792458 108 [m/s] 70 4 10 H/m

120 8 85418782 10 F/m.

= ck

= 2 f [rad/s]

f = 1/T [Hz]

, cosxE t z A t kz

= m fm

vacuum media

€

=1

μ rμ0ε rε 0

0 0k

€

k = ω μ rμ0ε rε 0

vacuummedium

ExampleExample

KFCC 1270 AM (1270 [kHz])

61.270 10 [Hz]f

/ 236.06 [m]c f

71/ 7.8740 10 [s] 0.78740 [ s]T f

62 7.979645 10 [rad/s]f

2 / 0.026617 [rad/m]k

6cos 7 979645 10 0 026617 [V/m]xE t,z A . t . z

Calculate all of the parameters and write down an expression for the electric field of this wave.