SINUSOIDAL WAVES LAB

Professor Ahmadiand Robert Proie

OBJECTIVES Learn to Mathematically Describe

Sinusoidal Waves Refresh Complex Number Concepts

DESCRIBING A SINUSOIDAL WAVE

SINUSOIDAL WAVES Described by the equation

Y = A ∙ sin(ωt + φ) A = Amplitude ω = Frequency in Radians (Angular Frequency) φ = Initial Phase

X=TIME (seconds)

Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Y = 5∙sin(2π∙0.05∙t + 0)

Y = 5 ∙ sin(2π∙0.05∙t+ 0)

SINUSOIDAL WAVES: AMPLITUDE

Definition: Vertical distance between peak value and center value.

X=TIME (seconds)

Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Amplitude = 5 units

SINUSOIDAL WAVES: PEAK TO PEAK VALUE

Definition: Vertical distance between the maximum and minimum peak values.

X=TIME (seconds)

Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Peak to Peak Value= 10 units

Y = 5 ∙ sin(2π∙0.05∙t+ 0)

SINUSOIDAL WAVES: FREQUENCY

Definition: Number of cycles that complete within a given time period.

Standard Unit: Hertz (Hz) 1 Hz = 1 cycle / second

For Sine Waves: Frequency = ω / (2π) Ex. (2π*0.05) / (2π) = 0.05 Hz

X=TIME (seconds)

Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Frequency = 0.05 cycles/secondOr

Frequency = 0.05 Hz

f= 1 / Tω = 2 π f

Y = 5 ∙ sin(2π∙0.05∙t+ 0)

SINUSOIDAL WAVES: PERIOD

Definition: Time/Duration from the beginning to the end of one cycle.

Standard Unit: seconds (s) For Sine Waves: Period = (2π) / ω

Ex. (2π) / (2π*0.05)= 20 seconds

X=TIME (seconds)

Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Period = 20 secondsf= 1 / Tω = 2 π f

SINUSOIDAL WAVES: PHASE Sinusoids do not always have a value of

0 at Time = 0.

Time (s)Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Time (s)

Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Time (s)Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Time (s)

Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

SINUSOIDAL WAVES: PHASE Phase indicates position of wave at

Time = 0 One full cycle takes 360º or 2π radians

(X radians) ∙ 180 / (2 π) = Y degrees (Y degrees ) ∙ (2 π) /180 = X radians

Phase can also be represented as an angle Often depicted as a vector within a circle of

radius 1, called a unit circle

Image from http://en.wikipedia.org/wiki/Phasor, Feb 2011

SINUSOIDAL WAVES: PHASE The value at Time = 0 determines the

phase.

Time (s)Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Time (s)

Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Phase = 0º or 0 radians Phase = 90º or π/2 radians

SINUSOIDAL WAVES: PHASE The value at Time = 0 determines the

phase.

Time (s)Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Time (s)

Ampl

itude

5 10 15 20

5

2.5

-2.5

-5

Phase = 180º or π radians Phase = 270º or 3π/4 radians

WORKING WITH COMPLEX NUMBERS

COMPLEX NUMBERS Commonly represented 2 ways

Rectangular form: z = a + bi a = real part b = imaginary part

Polar Form: z = r(cos(φ) + i sin(φ)) r = magnitude φ = phase

Given a & b

Given r & φ

a a r cos(φ)

b b r sin(φ)

r rφ φ

22 ba

ab1tan

Conversion Chart

a

b

φ

r

COMPLEX NUMBERS: EXAMPLE Given: 4.0 + 3.0i, convert to polar form.

1. r = (4.02 +3.02)(1/2) = 5.02. φ = 0.643. Solution: 5.0(cos(0.64) + i sin(0.64))

Given: 2.5(cos(.35) + i sin(0.35)), convert to rectangular form.1. a = 2.5 cos(0.35) = 2.32. b = 2.5 sin(0.35) = 0.863. Solution = 2.3 + 0.86i

COMPLEX NUMBERS: EULER’S FORMULA Polar form complex numbers are often

represented with exponentials using Euler’s Formula

e(iφ) = cos(φ) + i sin(φ)or

r*e(iφ) = r ∙ (cos(φ) + i sin(φ)) e is the base of the natural log, also called

Euler’s number or exponential.

COMPLEX NUMBERS: EULER’S FORMULA EXAMPLES Given: 4.0 + 3.0i, convert to polar

exponential form.1. r = (4.02 +3.02)(1/2) = 5.02. φ = 0.643. 5.0(cos(0.64) + i sin(0.64))4. Solution: 5.0e(0.64i)

Given: 2.5(cos(.35) + i sin(0.35)), convert to polar exponential form.1. Solution = 2.5e(0.35i)

PUTTING IT ALL TOGETHER: PHASOR INTRODUCTION

PHASOR INTRODUCTION We can use complex numbers and

Euler’s formula to represent sine and cosine waves.

We call this representation a phase vector or phasor.

Take the equation A ∙ cos(ωt + φ)Re{Aeiωteiφ}Re means Real Part

Convert to polar form

Re{Aeiφ}

Drop the frequency/ω term

Aφ

Drop the real part notationIMPORTANT: Common convention is to express phasors in terms of cosines as shown here.

Given: Express 5*sin(100t + 120°) in phasor notation.

PHASOR INTRODUCTION: EXAMPLES Given: Express 5*cos(100t + 30°) in

phasor notation.

Remember: sin(x) = cos(x-90°)

1. Re{5ei100tei30°}2. Re{5ei30°}3. Solution: 530°

1. 5*cos(100t + 30°)2. Re{5ei100tei30°}3. Re{5ei30°}4. Solution: 530°

4

3Vector representing phasor with magnitude 5 and 30°angle

Same solution!

LAB EXERCISES

SINUSOIDS: INSTRUCTIONS In the coming weeks, you will learn how

to measure alternating current (AC) signals using an oscilloscope. An interactive version of this tool is available at http://www.virtual-oscilloscope.com/simulati

on.html Using that simulator and the tips listed,

complete the exercises on the following slides.

Tip: Make sure you press the power button to turn on the simulated oscilloscope.

SINUSOIDS: INSTRUCTIONS For each problem, turn in a screenshot of the

oscilloscope and the answers to any questions asked.

Solutions should be prepared in a Word/Open Office document with at most one problem per page.

An important goal is to learn by doing, rather than simply copying a set of step-by-step instructions. Detailed instruction on using the simulator can be found athttp://www.virtual-oscilloscope.com/help/index.html and additional questions can be directed to your GTA.

PROBLEM 1: SINUSOIDS The display of an oscilloscope is

divided into a grid. Each line is called a division.

Vertical lines represent units of time.A. Which two cables produce signals a period closes to 8 ms?

B. What is the frequency of these signals?

C. What is the amplitude of these signals?

D. Capture an image of the oscilloscope displaying at least 1 cycle of each signal simultaneously.

Hint: You will need to use the “DUAL” button to display 2 signals at the same time.

PROBLEM 2: SINUSOIDS Horizontal lines represent units of

voltage.A. What is the amplitude of the pink cable’s signal? The orange cable?

B. What are their frequencies?C. What is the Peak-to-Peak voltage

of the sum of these two signals?D. Capture an image of the

oscilloscope displaying the addition of the pink and orange cables.

E. Repeat A-D for the pink and purple cables.

Hint: You will need to use the “ADD” button to add 2 signals together.

SINUSOIDS: INSTRUCTIONS Look at the image of the oscilloscope on

the following page and answer the questions.

PROBLEM 3: SINUSOIDS

A. What is the amplitude of the signal? What is the peak to peak voltage?

B. What is the frequency of the signal? What is the period.

C. What is the phase of the sine wave at time = 0? 0.5 V/ Div

0.5 ms / Div

Time = 0 Location

COMPLEX NUMBERS: INSTRUCTIONS For each of these problems, you must

include your work. Please follow the steps listed previously in the lecture.

PROBLEM 4: COMPLEX NUMBERS Convert the following to polar,

sinusoidal form.A. 5+3iB. 12.2+7iC. -3+2iD. 6-8iE. -3π/2-πiF. 2+17i

PROBLEM 5: COMPLEX NUMBERS Convert the following to rectangular

form.A. 1.8(cos(.35) + i sin(0.35))B. -3.5(cos(1.2) + i sin(1.2))C. 0.4(cos(-.18) + i sin(-.18))D. 3.8e(3.8i)

E. -2.4e(-15i)

F. 1.5e(12.2i)

PROBLEM 6: COMPLEX NUMBERS Convert the following to polar,

exponential form using Euler’s Formula.A. 1.8(cos(.35) + i sin(0.35))B. -3.5(cos(1.2) + i sin(1.2))C. 0.4(cos(-.18) + i sin(-.18))D. 6-8iE. -3π/2-πiF. 2+17i

PHASORS: INSTRUCTIONS For each of these problems, you must

include your work. Please follow the steps listed previously in the lecture.

PROBLEM 7: PHASORS Convert the following items into phasor

notation.A. 3.2*cos(15t+7°)B. -2.8*cos(πt-13°)C. 1.6*sin(2πt+53°)D. -2.8*sin(-t-128°)

PROBLEM 8: PHASORS Convert the following items from phasor

notation into its cosine equivalent. Express phases all values in radians where relavent.1. 530° with a frequency of 17 Hz2. -183127° with a frequency of 100 Hz3. 15-32° with a frequency of 32 Hz4. -2.672° with a frequency of 64 Hz