Spfirst l03 Ev

Copyright Monash University 2009

Signal Processing

First

Lecture3Phasor Addition

Theorem

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READING ASSIGNMENTS

ThisLecture: Chapter2,Section26

OtherReading: AppendixA:ComplexNumbers AppendixB:MATLAB NextLecture:startChapter3

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LECTURE OBJECTIVES

Phasors=ComplexAmplitude ComplexNumbersrepresent Sinusoids

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Develop the ABSTRACTION: Adding Sinusoids = Complex Addition PHASOR ADDITION THEOREM

tjjtj eAeXetz )()(


AVOID Trigonometry Algebra,evencomplex,isEASIER !!! Canyourecallcos(1+2)? Use:realpartofej12=cos(1+2)

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2121 )( jjj eee )sin)(cossin(cos 2211 jj

(...))sinsincos(cos 2121 j


Eulers FORMULA

ComplexExponential Realpartiscosine Imaginarypartissine Magnitudeisone

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)sin()cos( tjte tj )sin()cos( je j


Real & Imaginary Part Plots

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PHASE DIFFERENCE = /2


COMPLEX EXPONENTIAL

Interpretthisasa RotatingVector t Anglechangesvs.time ex:rad/s) Rotates in0.01secs

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e jj cos( ) sin( )

)sin()cos( tjte tj


Cos = REAL PART

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cos(t) e e j t Real Part of Eulers

x(t) Acos(t )General Sinusoid

A cos( t ) e Ae j ( t ) e Ae je j t

So,


COMPLEX AMPLITUDE

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x(t) Acos(t ) e Ae jej t General Sinusoid

z(t) Xe jt X Ae jComplex AMPLITUDE = X

x(t) e Xe j t e z(t) Sinusoid = REAL PART of (Aej)ejt


POP QUIZ: Complex Amp FindtheCOMPLEXAMPLITUDEfor:

UseEULERsFORMULA:

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5.03 jeX

)5.077cos(3)( ttx

tjjtj

eee

eetx

775.0

)5.077(

3

3)(


WANT to ADD SINUSOIDS ALLSINUSOIDShaveSAME FREQUENCY HOWtoGET{Amp,Phase} ofRESULT?

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ADD SINUSOIDS

SumSinusoidhasSAME Frequency

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PHASOR ADDITION RULE

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Get the new complex amplitude by complex addition


Phasor Addition Proof

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POP QUIZ: Add Sinusoids

ADDTHESE2SINUSOIDS:

COMPLEXADDITION:

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5.00 31 jj ee

)5.077cos(3)(

)77cos()(

2

1

ttx

ttx


POP QUIZ (answer)

COMPLEXADDITION:

CONVERTbacktocosineform:

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j 3 3e j 0.51

31 j 3/231 jej

)77cos(2)( 33 ttx


ADD SINUSOIDS EXAMPLE

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tm1

tm2

tm3

)()()( 213 txtxtx

)(1 tx

)(2 tx


Convert Time-Shift to Phase

Measurepeaktimes: tm1=0.0194,tm2=0.0556,tm3=0.0394

Converttophase (T=0.1) 1=tm1 =2(tm1 /T)=70/180, 2=200/180

Amplitudes A1=1.7,A2=1.9,A3=1.532

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Phasor Add: Numerical

ConvertPolartoCartesian X1 =0.5814+j1.597 X2 =1.785 j0.6498 sum= X3 =1.204+j0.9476

ConvertbacktoPolar X3=1.532atangle141.79/180 Thisisthesum

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ADD SINUSOIDS

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VECTOR(PHASOR)ADD

X1

X2

X3

Spfirst l03 Ev

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