ECE 1100: Introduction to Electrical and Computer Engineering Sinusoidal Signals Waves t v(t)v(t) Wanda Wosik Associate Professor, ECE Dept. Spring 2011 Slides developed by Dr. Jackson

Basic 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 Waveform A = amplitude of sinusoidal waveform  = “radian frequency” of sinusoidal waveform [radians/s]  = phase of sinusoidal signal [radians] t v (t)v (t) A -A 

Period 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)v (t) T

Frequency 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)v (t) [s] 1.5

Radian Frequency Since the cosine function repeats after 2 , we have t v (t)v (t) T  = 2  f [rad/s] Hence: or  Rotation with angular frequency 

Summary t v(t)v(t) A -A  = 2  f [rad/s] f = 1/T [Hz]

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

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

z [m] h (z)h (z) crest trough v = velocityWavenumber Assumed form of wave: k is “wavenumber” of wave. Wavelength z [m] h(z)h(z) The wavelength is the distance it takes for the waveform to repeat (for a fixed time).

Wave at Fixed Observation Point Next question: what is the velocity that the observer will feel? z [m] h (z)h (z) v = velocity observer zz Point on crest µ 0 =4  x10 -7 H/m  0 =8.85x F/m c=2.997x10 8 m/s v = velocity

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 = z 0 ? z [m] h (z)h (z) v = velocity observer We can pretend that the wave is fixed and the observer is moving backwards at velocity v.

Wave at Fixed Observation Point (cont.) z [m] h (z)h (z) v = velocity observer

Wave at Fixed Observation Point (cont.) z [m] h (z)h (z) v = velocity observer

Define Wave at Fixed Observation Point (cont.) The observed amplitude varies sinusoidally in time!

General Form of Wave z [m] h (t, z)h (t, z) v = velocity Allowing for both t and z to be arbitrary, we have.

General Form of Wave (cont.) For the velocity we can also write

E H Electromagnetic Waves Wave propagation with speed of light c

Electromagnetic 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 velocity = c (speed of light) The power flows in the direction E  H

Electromagnetic Wave (cont.) z c = speed of light electric 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.) z earth x power flow ( z direction) The electric field vector is in the direction of the transmitting antenna.

Transmitting Antenna AM Radio: 550 kHz < f < 1610 kHz ExEx Electric field is vertically polarized z earth x

Transmitting Antenna (cont.) FM Radio: 88 MHz < f < 108 MHz Electric field is horizontally polarized z EyEy earth x VHF TV: MHz < f < 216 MHz UHF TV: 470 MHz < f < 806 MHz

Equation for Electric Field z c = speed of light electric field vector x Note: It is the electric field that would be received by a wire antenna.

Value of k From Maxwell’s equations, it can be shown that: (permeability of free space) (permittivity of free space)

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 c =  10 8 [m/s] (exact defined quantity)

Summary of Wave Formulas k = 2  / c = f c =  10 8 [m/s]  = ck  = 2  f [rad/s] f = 1/T [Hz] = m f m vacuum media vacuum  medium

Example KFCC 1270 AM (1270 [kHz]) Calculate all of the parameters and write down an expression for the electric field of this wave.