sinusoidal waves lab
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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
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