Transcript
Page 1: LECTURE 4 # Ch2 Phasor Addition Theorem...DSP-First, 2/e LECTURE 4 # Ch2 Phasor Addition Theorem Aug 2016 © 2003-2016, JH McClellan & RW Schafer 1 MODIFIED TLH ADDING PHASORS WITH

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

Page 2: LECTURE 4 # Ch2 Phasor Addition Theorem...DSP-First, 2/e LECTURE 4 # Ch2 Phasor Addition Theorem Aug 2016 © 2003-2016, JH McClellan & RW Schafer 1 MODIFIED TLH ADDING PHASORS WITH

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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)cos(

)cos()(

0

1

0

tA

tAtxN

k

kk

jN

k

j

k AeeA k

1

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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 4

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

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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 6

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 7

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 8

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 9

WANT to ADD SINUSOIDS

Main point to remember: Adding

sinusoids of common frequency results in

sinusoid with SAME frequency

Page 10: LECTURE 4 # Ch2 Phasor Addition Theorem...DSP-First, 2/e LECTURE 4 # Ch2 Phasor Addition Theorem Aug 2016 © 2003-2016, JH McClellan & RW Schafer 1 MODIFIED TLH ADDING PHASORS WITH

)cos(

)cos()(

0

1

0

tA

tAtxN

k

kk

jN

k

j

k AeeA k

1

Aug 2016 © 2003-2016, JH McClellan & RW Schafer 10

PHASOR ADDITION RULE

Get the new complex amplitude by complex addition

Page 29

Page 11: LECTURE 4 # Ch2 Phasor Addition Theorem...DSP-First, 2/e LECTURE 4 # Ch2 Phasor Addition Theorem Aug 2016 © 2003-2016, JH McClellan & RW Schafer 1 MODIFIED TLH ADDING PHASORS WITH

)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

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Phasor Addition Proof

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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 12

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 13

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 14

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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Page 16: LECTURE 4 # Ch2 Phasor Addition Theorem...DSP-First, 2/e LECTURE 4 # Ch2 Phasor Addition Theorem Aug 2016 © 2003-2016, JH McClellan & RW Schafer 1 MODIFIED TLH ADDING PHASORS WITH

Can we do these steps?

Sure but MATLAB saves us!

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% 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!

Page 17: LECTURE 4 # Ch2 Phasor Addition Theorem...DSP-First, 2/e LECTURE 4 # Ch2 Phasor Addition Theorem Aug 2016 © 2003-2016, JH McClellan & RW Schafer 1 MODIFIED TLH ADDING PHASORS WITH

Convert Sinusoids to Phasors

Each sinusoid Complex Amp

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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 18

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 19

ADDING SINUSOIDS IS

COMPLEX ADDITION

VECTOR(PHASOR)ADD

X1

X2

X3

Page 20: LECTURE 4 # Ch2 Phasor Addition Theorem...DSP-First, 2/e LECTURE 4 # Ch2 Phasor Addition Theorem Aug 2016 © 2003-2016, JH McClellan & RW Schafer 1 MODIFIED TLH ADDING PHASORS WITH

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.

Page 21: LECTURE 4 # Ch2 Phasor Addition Theorem...DSP-First, 2/e LECTURE 4 # Ch2 Phasor Addition Theorem Aug 2016 © 2003-2016, JH McClellan & RW Schafer 1 MODIFIED TLH ADDING PHASORS WITH

Copyright © 2016, 1998 Pearson Education, Inc. All Rights Reserved

Table 2-3: Phasor Addition Example


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