lecture 4 # ch2 phasor addition theorem% convert phasor to rectangular format short...
TRANSCRIPT
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DSP-First, 2/e
LECTURE 4 # Ch2
Phasor Addition Theorem
Aug 2016 1© 2003-2016, JH McClellan & RW Schafer
MODIFIED TLH
ADDING PHASORS WITH THE SAME FREQUENCY
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 2
READING ASSIGNMENTS
▪ This Lecture:
▪ Chapter 2, Section 2-6
▪ Other Reading:
▪ Appendix A: Complex Numbers
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2020 © 2003-2016, JH McClellan & RW Schafer 3
Dr. Van Veen Strikes Again
3. Complex Numbers Review (Wouldn’t hurt to review) 10:22Review of how to work with complex numbers in rectangular and polar coordinates.
https://www.youtube.com/watch?v=UAn9uah7puU&list=PLGI7M8vwfrFNO-gQ1xoJmN3bJy2-wp2J3
4. Complex Sinusoid Representations for Real Sinusoids 13:21
https://www.youtube.com/watch?v=Tm3gI6PQOYo&feature=youtu.be
https://www.youtube.com/watch?v=UAn9uah7puU&list=PLGI7M8vwfrFNO-gQ1xoJmN3bJy2-wp2J3https://www.youtube.com/watch?v=Tm3gI6PQOYo&feature=youtu.be
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)cos(
)cos()(
0
1
0
+=
+==
tA
tAtxN
k
kk
jN
k
j
k AeeAk
=
=1
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 4
PHASOR ADDITION RULE
Get the new complex amplitude by complex addition
Page 29
Find Amplitude and
Phase – ω is Known!
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 5
LECTURE OBJECTIVES
▪ Phasors = Complex Amplitude
▪ Complex Numbers represent Sinusoids
▪ Develop the ABSTRACTION:
▪ Adding Sinusoids = Complex Addition
▪ PHASOR ADDITION THEOREM
}){()cos( tjj eAetA =+
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Adding Complex Numbers
▪ Polar Form
▪ Could convert to Cartesian and back out
▪ Use MATLAB
▪ Visualize the vectors
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 6
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 7
Cos = REAL PART
}{
}{)cos( )(
tjj
tj
eAe
AetA
=
=+ +
What about sinusoidal signals over time?Real part of Euler’s
}{)cos( tjet =
General Sinusoid
Complex Amplitude: Constant Varies with time
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 8
POP QUIZ: Complex Amp
▪ Find the COMPLEX AMPLITUDE for:
▪ Use EULER’s FORMULA:
5.03 jeX =
)5.077cos(3)( += ttx
}3{
}3{)(
775.0
)5.077(
tjj
tj
ee
etx
=
= +
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 9
POP QUIZ-2: Complex Amp
▪ Determine the 60-Hz sinusoid whose
COMPLEX AMPLITUDE is:
▪ Convert X to POLAR:
33 jX +=
)3/120cos(12)( += ttx
}12{
})33{()(
1203/
)120(
tjj
tj
ee
ejtx
=
+=
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 10
WANT to ADD SINUSOIDS
▪ Main point to remember: Adding
sinusoids of common frequency results in
sinusoid with SAME frequency
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)cos(
)cos()(
0
1
0
+=
+==
tA
tAtxN
k
kk
jN
k
j
k AeeAk
=
=1
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 11
PHASOR ADDITION RULE
Get the new complex amplitude by complex addition
Page 29
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( ) )cos(
}{)cos(
0
1
1
1
)(
1
0
0
0
0
0
+==
=
=
=+
=
=
=
+
=
tAeAe
eeA
eeA
eAtA
tjj
tjN
k
j
k
N
k
tjj
k
N
k
tj
k
N
k
kk
k
k
k
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 12
Phasor Addition Proof
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 13
POP QUIZ: Add Sinusoids
▪ ADD THESE 2 SINUSOIDS:
▪ COMPLEX (PHASOR) ADDITION:
5.031 jj ee +−
)5.077cos(3)(
)77cos()(
2
1
+=
−=
ttx
ttx
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 14
POP QUIZ (answer)
▪ COMPLEX ADDITION:
▪ CONVERT back to cosine form:
)77cos(2)(3
23
+= ttx
1−=− je
3/2231 jej =+−
5.031 jj ee +−
33 2/ je j =
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 15
ADD SINUSOIDS EXAMPLE
▪ ALL SINUSOIDS have SAME FREQUENCY
▪ HOW to GET {Amp,Phase} of RESULT ?
}{
)cos()()()(
20
213
tjj eAe
tAtxtxtx
=
+=+=
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 16
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Can we do these steps?
Sure but MATLAB saves us!
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 17
% Convert Phasor to rectangular
format short
x1=1.7*exp(j*70*pi/180)
% x1= 0.5814+ 1.5975i
x2=1.9*exp(j*200*pi/180)
% x2 = -1.7854 - 0.6498i
x3=x1+x2 % x3 = -1.2040 + 0.9476i
% Convert x3 to polar
magx3=abs(x3) % magx3 = 1.5322
x3theta=angle(x3) % x3theta = 2.4748 rad
thetadeg=x3theta*180/pi
% thetadeg = 141.7942 degrees
Piece of Cake!
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Convert Sinusoids to Phasors
▪ Each sinusoid → Complex Amp
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 18
180/200
180/70
9.1)180/20020cos(9.1
7.1)180/7020cos(7.1
j
j
et
et
→+
→+
)180/79.14120cos(532.1
532.1
?9.17.1
180/79.141
180/200180/70
+→
=+
t
e
ee
j
jj
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 19
Phasor Add: Numerical
▪ Convert Polar to Cartesian
▪ X1 = 0.5814 + j1.597
▪ X2 = -1.785 - j0.6498
▪ sum =
▪ X3 = -1.204 + j0.9476
▪ Convert back to Polar▪ X3 = 1.532 at angle 141.79/180
▪ This is the sum
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 20
ADDING SINUSOIDS IS
COMPLEX ADDITION
VECTOR(PHASOR)ADD
X1
X2
X3
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Copyright © 2016, 1998 Pearson Education, Inc. All Rights Reserved
Phasor Addition Rule: Example
Figure 2-15: (a) Adding sinusoids by doing a phasor addition, which is actually a graphical
vector sum. (b) The time of the signal maximum is marked on each ( )tX j plot.
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Copyright © 2016, 1998 Pearson Education, Inc. All Rights Reserved
Table 2-3: Phasor Addition Example