Sinusoidal Waves Lab

1. Sinusoidal Waves Lab Professor Ahmadi and Robert Proie

2. Objectives • Learn to Mathematically Describe Sinusoidal Waves • Refresh Complex Number Concepts

3. Describing a Sinusoidal Wave

4. Sinusoidal Waves • Described by the equation • Y = A ∙ sin(ωt + φ) • A = Amplitude • ω = Frequency in Radians (Angular Frequency) • φ = Initial Phase 5 2.5 -2.5 -5 Y = 5∙sin(2π∙0.05∙t + 0) Amplitude X=TIME (seconds) 5 10 15 20

5. Sinusoidal Waves: Amplitude 5 2.5 -2.5 -5 • Definition: Vertical distance between peak value and center value. Y = 5∙ sin(2π∙0.05∙t+ 0) Amplitude X=TIME (seconds) 5 10 15 20 Amplitude = 5 units

6. Sinusoidal Waves: Peak to Peak Value 5 2.5 -2.5 -5 • Definition: Vertical distance between the maximum and minimum peak values. Amplitude X=TIME (seconds) 5 10 15 20 Peak to Peak Value= 10 units

7. Sinusoidal Waves: Frequency Y = 5∙ sin(2π∙0.05∙t+ 0) 5 2.5 -2.5 -5 • 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 Amplitude X=TIME (seconds) 5 10 15 20 f= 1 / T ω= 2 π f Frequency = 0.05 cycles/second Or Frequency = 0.05 Hz

8. Sinusoidal Waves: Period 5 2.5 -2.5 -5 • 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 Y = 5 ∙ sin(2π∙0.05∙t+ 0) Amplitude X=TIME (seconds) 5 10 15 20 f= 1 / T ω= 2 π f Period = 20 seconds

9. Sinusoidal Waves: Phase • Sinusoids do not always have a value of 0 at Time = 0. 5 2.5 -2.5 -5 5 2.5 -2.5 -5 5 2.5 -2.5 -5 5 2.5 -2.5 -5 Time (s) Time (s) Amplitude Amplitude Amplitude Amplitude 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 Time (s) Time (s)

10. 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

11. Sinusoidal Waves: Phase • The value at Time = 0 determines the phase. 5 2.5 -2.5 -5 5 2.5 -2.5 -5 Time (s) Amplitude Amplitude 5 10 15 20 5 10 15 20 Time (s) Phase = 0º or 0 radians Phase = 90º or π/2radians

12. Sinusoidal Waves: Phase • The value at Time = 0 determines the phase. 5 2.5 -2.5 -5 5 2.5 -2.5 -5 Time (s) Amplitude Amplitude Phase = 180º or π radians Phase = 270º or 3π/4 radians 5 10 15 20 5 10 15 20 Time (s)

13. Working with Complex Numbers

14. 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 r b a φ Conversion Chart

15. Complex Numbers: Example • Given: 4.0 + 3.0i, convert to polar form. • r = (4.02 +3.02)(1/2) = 5.0 • φ = 0.64 • Solution: 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) = 2.3 • b = 2.5 sin(0.35) = 0.86 • Solution = 2.3 + 0.86i

16. Complex Numbers: Euler’s Formula • Polar form complex numbers are often represented with exponentials using Euler’s Formula e(iφ) = cos(φ) + isin(φ)or r*e(iφ) = r ∙ (cos(φ) + isin(φ)) • e is the base of the natural log, also called Euler’s number or exponential.

17. Complex Numbers: Euler’s Formula Examples • Given: 4.0 + 3.0i, convert to polar exponential form. • r = (4.02 +3.02)(1/2) = 5.0 • φ = 0.64 • 5.0(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)

18. Putting it All Together: Phasor Introduction

19. 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) Convert to polar form Re means Real Part Re{Aeiωteiφ} Drop the frequency/ω term Re{Aeiφ} IMPORTANT: Common convention is to express phasors in terms of cosines as shown here. Drop the real part notation Aφ

20. Phasor Introduction: Examples • Given: Express 5*cos(100t + 30°) in phasor notation. Vector representing phasor with magnitude 5 and 30°angle 3 • Re{5ei100tei30°} • Re{5ei30°} • Solution: 530° 4 • Given: Express 5*sin(100t + 120°) in phasor notation. • 5*cos(100t + 30°) • Re{5ei100tei30°} • Re{5ei30°} • Solution: 530° Remember: sin(x) = cos(x-90°) Same solution!

21. Lab Exercises

22. 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/simulation.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.

23. 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.

24. 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. • 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.

25. Problem 2: Sinusoids • Horizontal lines represent units of voltage. • What is the amplitude of the pink cable’s 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.

26. Sinusoids: Instructions • Look at the image of the oscilloscope on the following page and answer the questions.

27. Problem 3: Sinusoids • What 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 ms / Div 0.5 V/ Div Time = 0 Location

28. Complex Numbers: Instructions • For each of these problems, you must include your work. Please follow the steps listed previously in the lecture.

29. Problem 4: Complex Numbers • Convert the following to polar, sinusoidal form. • 5+3i • 12.2+7i • -3+2i • 6-8i • -3π/2-πi • 2+17i

30. Problem 5: Complex Numbers • Convert 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)

31. Problem 6: Complex Numbers • Convert the following to polar, exponential form using Euler’s 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 • -3π/2-πi • 2+17i

32. Phasors: Instructions • For each of these problems, you must include your work. Please follow the steps listed previously in the lecture.

33. Problem 7: Phasors • Convert the following items into phasor notation. • 3.2*cos(15t+7°) • -2.8*cos(πt-13°) • 1.6*sin(2πt+53°) • -2.8*sin(-t-128°)

34. Problem 8: Phasors • Convert the following items from phasor notation into its cosine equivalent. Express phases all values in radians where relavent. • 530° with a frequency of 17 Hz • -183127° with a frequency of 100 Hz • 15-32° with a frequency of 32 Hz • -2.672° with a frequency of 64 Hz