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Sinusoidal Waves Lab. Professor Ahmadi and 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( ω t + φ ) A = Amplitude - PowerPoint PPT Presentation

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Sinusoidal Waves LabProfessor Ahmadiand Robert ProieObjectivesLearn to Mathematically Describe Sinusoidal WavesRefresh Complex Number ConceptsDescribing a Sinusoidal WaveSinusoidal WavesDescribed by the equationY = A sin(t + )A = Amplitude = Frequency in Radians (Angular Frequency) = Initial Phase

X=TIME (seconds)Amplitude5 10 15 205

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-5Y = 5sin(20.05t + 0)Y = 5 sin(20.05t+ 0)Sinusoidal Waves: AmplitudeDefinition: Vertical distance between peak value and center value.X=TIME (seconds)Amplitude5 10 15 205

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-5Amplitude = 5 unitsSinusoidal Waves: Peak to Peak ValueDefinition: Vertical distance between the maximum and minimum peak values.X=TIME (seconds)Amplitude5 10 15 205

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-5Peak to Peak Value= 10 unitsY = 5 sin(20.05t+ 0)Sinusoidal Waves: FrequencyDefinition: Number of cycles that complete within a given time period.Standard Unit: Hertz (Hz)1 Hz = 1 cycle / secondFor Sine Waves: Frequency = / (2)Ex. (2*0.05) / (2) = 0.05 HzX=TIME (seconds)Amplitude5 10 15 205

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-5Frequency = 0.05 cycles/secondOrFrequency = 0.05 Hzf= 1 / T = 2 fY = 5 sin(20.05t+ 0)Sinusoidal Waves: PeriodDefinition: Time/Duration from the beginning to the end of one cycle.Standard Unit: seconds (s)For Sine Waves: Period = (2) / Ex. (2) / (2*0.05)= 20 seconds

X=TIME (seconds)Amplitude5 10 15 205

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-5Period = 20 secondsf= 1 / T = 2 fSinusoidal Waves: PhaseSinusoids do not always have a value of 0 at Time = 0.Time (s)Amplitude5 10 15 205

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-5Time (s)Amplitude5 10 15 205

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Sinusoidal Waves: PhasePhase indicates position of wave at Time = 0One full cycle takes 360 or 2 radians(X radians) 180 / (2 ) = Y degrees(Y degrees ) (2 ) /180 = X radiansPhase can also be represented as an angleOften depicted as a vector within a circle of radius 1, called a unit circle

Image from http://en.wikipedia.org/wiki/Phasor, Feb 2011Sinusoidal Waves: PhaseThe value at Time = 0 determines the phase.Time (s)Amplitude5 10 15 205

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Time (s)Amplitude5 10 15 205

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-5Phase = 0 or 0 radiansPhase = 90 or /2 radiansSinusoidal Waves: PhaseThe value at Time = 0 determines the phase.Time (s)Amplitude5 10 15 205

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Phase = 180 or radiansPhase = 270 or 3/4 radiansWorking with Complex NumbersComplex NumbersCommonly represented 2 waysRectangular form: z = a + bia = real partb = imaginary partPolar Form: z = r(cos() + i sin())r = magnitude = phase

Given a & bGiven r & aar cos()bbr sin()rr

Conversion ChartabrComplex Numbers: ExampleGiven: 4.0 + 3.0i, convert to polar form.r = (4.02 +3.02)(1/2) = 5.0 = 0.64Solution: 5.0(cos(0.64) + i sin(0.64))Given: 2.5(cos(.35) + i sin(0.35)), convert to rectangular form.a = 2.5 cos(0.35) = 2.3b = 2.5 sin(0.35) = 0.86Solution = 2.3 + 0.86iComplex Numbers: Eulers FormulaPolar form complex numbers are often represented with exponentials using Eulers Formulae(i) = cos() + i sin()orr*e(i) = r (cos() + i sin())e is the base of the natural log, also called Eulers number or exponential.

Complex Numbers: Eulers Formula ExamplesGiven: 4.0 + 3.0i, convert to polar exponential form.r = (4.02 +3.02)(1/2) = 5.0 = 0.645.0(cos(0.64) + i sin(0.64))Solution: 5.0e(0.64i)

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

Putting it All Together: Phasor IntroductionPhasor IntroductionWe can use complex numbers and Eulers formula to represent sine and cosine waves.We call this representation a phase vector or phasor.Take the equation A cos(t + )

Re{Aeitei}Re means Real PartConvert to polar formRe{Aei}Drop the frequency/ term ADrop 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: ExamplesGiven: Express 5*cos(100t + 30) in phasor notation.Remember: sin(x) = cos(x-90)Re{5ei100tei30}Re{5ei30}Solution: 5305*cos(100t + 30)Re{5ei100tei30}Re{5ei30}Solution: 53043Vector representing phasor with magnitude 5 and 30angleSame solution!Lab ExercisesSinusoids: InstructionsIn 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/simulation.htmlUsing 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: InstructionsFor 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: SinusoidsThe display of an oscilloscope is divided into a grid. Each line is called a division.Vertical lines represent units of time.

Which two cables produce signals a period closes to 8 ms?What is the frequency of these signals?What is the amplitude of these signals?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: SinusoidsHorizontal lines represent units of voltage.

What is the amplitude of the pink cables signal? The orange cable?What are their frequencies?What is the Peak-to-Peak voltage of the sum of these two signals?Capture an image of the oscilloscope displaying the addition of the pink and orange cables.Repeat A-D for the pink and purple cables.

Hint: You will need to use the ADD button to add 2 signals together.Sinusoids: InstructionsLook at the image of the oscilloscope on the following page and answer the questions.Problem 3: SinusoidsWhat is the amplitude of the signal? What is the peak to peak voltage?What is the frequency of the signal? What is the period.What is the phase of the sine wave at time = 0?

0.5 V/ Div0.5 ms / DivTime = 0 LocationComplex Numbers: InstructionsFor each of these problems, you must include your work. Please follow the steps listed previously in the lecture. Problem 4: Complex NumbersConvert the following to polar, sinusoidal form.5+3i12.2+7i-3+2i6-8i-3/2-i2+17iProblem 5: Complex NumbersConvert the following to rectangular form.1.8(cos(.35) + i sin(0.35))-3.5(cos(1.2) + i sin(1.2))0.4(cos(-.18) + i sin(-.18))3.8e(3.8i)-2.4e(-15i)1.5e(12.2i)

Problem 6: Complex NumbersConvert the following to polar, exponential form using Eulers Formula.1.8(cos(.35) + i sin(0.35))-3.5(cos(1.2) + i sin(1.2))0.4(cos(-.18) + i sin(-.18))6-8i-3/2-i2+17i

Phasors: InstructionsFor each of these problems, you must include your work. Please follow the steps listed previously in the lecture.Problem 7: PhasorsConvert the following items into phasor notation.3.2*cos(15t+7)-2.8*cos(t-13)1.6*sin(2t+53)-2.8*sin(-t-128)Problem 8: PhasorsConvert the following items from phasor notation into its cosine equivalent. Express phases all values in radians where relavent.530 with a frequency of 17 Hz-183127 with a frequency of 100 Hz15-32 with a frequency of 32 Hz-2.672 with a frequency of 64 Hz