Professor Ahmadi and Robert Proie 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(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

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

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

Sinusoidal Waves: Frequency 5 2.5 -2.5 -5 Y = 5 ∙ sin(2π∙0.05∙t+ 0) 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 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

Sinusoidal Waves: Period 5 2.5 -2.5 -5 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 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

Sinusoidal Waves: Phase Sinusoids do not always have a value of 0 at Time = 0. Time (s) Amplitude 5 10 15 20 5 2.5 -2.5 -5 Time (s) Amplitude 5 10 15 20 5 2.5 -2.5 -5 Time (s) Amplitude 5 10 15 20 5 2.5 -2.5 -5 Time (s) Amplitude 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) Amplitude 5 10 15 20 5 2.5 -2.5 -5 Time (s) Amplitude 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. Phase = 180º or π radians Phase = 270º or 3π/4 radians Time (s) Amplitude 5 10 15 20 5 2.5 -2.5 -5 Time (s) Amplitude 5 10 15 20 5 2.5 -2.5 -5

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 r b a φ Conversion Chart Given a & b Given r & φ a r cos(φ) b r sin(φ) r φ

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

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. 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)

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) 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

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!

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/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.

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 at http://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. 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: 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.

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

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

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. 5+3i 12.2+7i -3+2i 6-8i -3π/2-πi 2+17i

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)

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

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. 3.2*cos(15t+7°) -2.8*cos(πt-13°) 1.6*sin(2πt+53°) -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. 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