Transcript EET 159 PowerPoint Slides

EGR 1101 Unit 6
Sinusoids in Engineering
(Chapter 6 of Rattan/Klingbeil text)
Periodic Waveforms
Often the graph of a physical quantity (such as
position, velocity, voltage, current, etc.) versus time
repeats itself. We call this a periodic waveform.
Common shapes for periodic waveforms include:
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Square
Triangle
Sawtooth
Sinusoidal
See diagram at bottom of page:
http://en.wikipedia.org/wiki/Sinusoid
Sinusoids are the most important of these.
Sinusoids
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A sinusoid is a sine wave or a cosine
wave or any wave with the same shape,
shifted to the left or right.
Sinusoids arise in many areas of
engineering and science. We’ll look at
three areas:
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Circular motion
Simple harmonic motion
Alternating current
Amplitude, Frequency, Phase
Angle
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Any two sinusoids must have the same
shape, but can vary in three ways:
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Amplitude (height)
Frequency (how fast the values change)
Phase angle (how far shifted to the left or right)
We’ll use mathematical expressions for
sinusoids that specify these three
parameters. Example:
v(t) = 20 sin(180t + 30) V
Today’s Examples
1.
2.
3.
One-link robot in motion
Simple harmonic motion of a spring-mass
system
Adding sinusoids in an RL circuit
One Question, Three Answers
Three equivalent answers to the
question, “How fast is the robot arm
spinning?”
1. Period, T, unit = seconds (s)
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2.
Frequency, f, unit = hertz (Hz)
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Tells how many seconds for one revolution
Tells how many revolutions per second
Angular frequency, , unit = rad/s
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Tells size of angle covered per second
Relating T, f, and 
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If you know any one of these three
(period, frequency, angular
frequency), you can easily compute
the other two.
T = 1/f
 = 2f = 2/T
General Form of a Sinusoid
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The general form of a sinusoid is
v(t) = A sin (t + )
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where A is the amplitude,  is the
angular frequency, and  is the
phase angle.
Often  is given in degrees; you
must convert it to radians for
calculations.
Adding Sinusoids
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Many problems require us to find the
sum of two or more sinusoids.
A unique property of sinusoids: the
sum of sinusoids of the same
frequency is always another
sinusoid of that frequency.
You can’t make the same statement
for triangle waves, square waves,
sawtooth waves, or other waveshapes.
Adding Sinusoids (Continued)
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For example, if we add
10 sin (200t + 30) and
12 sin (200t + 45)
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we’ll get another sinusoid of the same
angular frequency, 200 rad/s.
But how do we figure out the
amplitude and phase angle of the
resulting sinusoid?
Adding Sinusoids (Continued)
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Our technique for adding sinusoids
relies heavily on these trig identities:
sin(x + y) = sin x cos y + cos x
sin(x  y) = sin x cos y  cos x
and
cos(x + y) = cos x cos y  sin x
cos(x  y) = cos x cos y + sin x
sin y
sin y
sin y
sin y