Spfirst l03 Ev
Transcript of Spfirst l03 Ev
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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 )()(
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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
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Eulers FORMULA
ComplexExponential Realpartiscosine Imaginarypartissine Magnitudeisone
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)sin()cos( tjte tj )sin()cos( je j
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Real & Imaginary Part Plots
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PHASE DIFFERENCE = /2
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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
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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,
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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
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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)(
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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
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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
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POP QUIZ (answer)
COMPLEXADDITION:
CONVERTbacktocosineform:
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j 3 3e j 0.51
31 j 3/231 jej
)77cos(2)( 33 ttx
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ADD SINUSOIDS EXAMPLE
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tm1
tm2
tm3
)()()( 213 txtxtx
)(1 tx
)(2 tx
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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