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Professor Ahmadi and Robert Proie

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Learn to Mathematically Describe Sinusoidal Waves Refresh Complex Number Concepts

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Described by the equation Y = A ∙ sin( ωt + φ) A = Amplitude ω = Frequency in Radians (Angular Frequency) φ = Initial Phase X=TIME (seconds) Amplitude Y = 5∙sin(2 π ∙ 0.05 ∙ t + 0)

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Definition: Vertical distance between peak value and center value. X=TIME (seconds) Amplitude Amplitude = 5 units

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Definition: Vertical distance between the maximum and minimum peak values. X=TIME (seconds) Amplitude Peak to Peak Value= 10 units

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Y = 5 ∙ sin(2 π ∙ 0.05 ∙ t+ 0) 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) Amplitude Frequency = 0.05 cycles/second Or Frequency = 0.05 Hz f= 1 / T ω = 2 π f

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Y = 5 ∙ sin(2 π ∙ 0.05 ∙ t+ 0) 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) Amplitude Period = 20 seconds f= 1 / T ω = 2 π f

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Sinusoids do not always have a value of 0 at Time = 0. Time (s) Amplitude Time (s) Amplitude Time (s) Amplitude Time (s) Amplitude

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Phase indicates position of wave at Time = 0 One full cycle takes 360 º 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 Feb 2011

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The value at Time = 0 determines the phase. Time (s) Amplitude Time (s) Amplitude Phase = 0 º or 0 radians Phase = 90º or π/2 radians

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The value at Time = 0 determines the phase. Time (s) Amplitude Time (s) Amplitude Phase = 180 º or π radians Phase = 270 º or 3π/4 radians

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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 & bGiven r & φ aar cos( φ) bbr sin( φ) rr φφ Conversion Chart a b φ r

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Given: i, convert to polar form. 1. r = ( ) (1/2) = φ = 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) = b = 2.5 sin(0.35) = Solution = i

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

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Given: i, convert to polar exponential form. 1. r = ( ) (1/2) = φ = (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)

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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{Ae i ωt e iφ } Re means Real Part Convert to polar form Re{A e iφ } Drop the frequency/ ω term AφAφ Drop the real part notation IMPORTANT: Common convention is to express phasors in terms of cosines as shown here.

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Given: Express 5*sin(100t + 120°) in phasor notation. Given: Express 5*cos(100t + 30°) in phasor notation. Remember: sin(x) = cos(x-90°) 1. Re{5e i100t e i30° } 2. Re{5e i30° } 3. Solution: 5 30 ° 1. 5*cos(100t + 30°) 2. Re{5e i100 te i30° } 3. Re{5e i30° } 4. Solution: 5 30° 4 3 Vector representing phasor with magnitude 5 and 30°angle Same solution!

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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 oscilloscope.com/simulation.html 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.

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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 oscilloscope.com/help/index.html and additional questions can be directed to your GTA.

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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.

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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.

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

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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

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

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Convert the following to polar, sinusoidal form. A. 5+3i B i C. -3+2i D. 6-8i E. -3π/2-πi F. 2+17i

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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)

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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. 6-8i E. -3π/2-πi F. 2+17i

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

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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°)

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Convert the following items from phasor notation into its cosine equivalent. Express phases all values in radians where relavent 30° with a frequency of 17 Hz 127° with a frequency of 100 Hz -32° with a frequency of 32 Hz 72° with a frequency of 64 Hz

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