Post on 02-Oct-2015
Copyright Monash University 2009
Signal Processing
First
Lecture3Phasor Addition
Theorem
1
Copyright Monash University 2009
READING ASSIGNMENTS
ThisLecture: Chapter2,Section26
OtherReading: AppendixA:ComplexNumbers AppendixB:MATLAB NextLecture:startChapter3
2
Copyright Monash University 2009
LECTURE OBJECTIVES
Phasors=ComplexAmplitude ComplexNumbersrepresent Sinusoids
3
Develop the ABSTRACTION: Adding Sinusoids = Complex Addition PHASOR ADDITION THEOREM
tjjtj eAeXetz )()(
Copyright Monash University 2009
AVOID Trigonometry Algebra,evencomplex,isEASIER !!! Canyourecallcos(1+2)? Use:realpartofej12=cos(1+2)
4
2121 )( jjj eee )sin)(cossin(cos 2211 jj
(...))sinsincos(cos 2121 j
Copyright Monash University 2009
Eulers FORMULA
ComplexExponential Realpartiscosine Imaginarypartissine Magnitudeisone
5
)sin()cos( tjte tj )sin()cos( je j
Copyright Monash University 2009
Real & Imaginary Part Plots
6
PHASE DIFFERENCE = /2
Copyright Monash University 2009
COMPLEX EXPONENTIAL
Interpretthisasa RotatingVector t Anglechangesvs.time ex:rad/s) Rotates in0.01secs
7
e jj cos( ) sin( )
)sin()cos( tjte tj
Copyright Monash University 2009
Cos = REAL PART
8
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,
Copyright Monash University 2009
COMPLEX AMPLITUDE
9
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
Copyright Monash University 2009
POP QUIZ: Complex Amp FindtheCOMPLEXAMPLITUDEfor:
UseEULERsFORMULA:
10
5.03 jeX
)5.077cos(3)( ttx
tjjtj
eee
eetx
775.0
)5.077(
3
3)(
Copyright Monash University 2009
WANT to ADD SINUSOIDS ALLSINUSOIDShaveSAME FREQUENCY HOWtoGET{Amp,Phase} ofRESULT?
11
Copyright Monash University 2009
ADD SINUSOIDS
SumSinusoidhasSAME Frequency
12
Copyright Monash University 2009
PHASOR ADDITION RULE
13
Get the new complex amplitude by complex addition
Copyright Monash University 2009
Phasor Addition Proof
14
Copyright Monash University 2009
POP QUIZ: Add Sinusoids
ADDTHESE2SINUSOIDS:
COMPLEXADDITION:
15
5.00 31 jj ee
)5.077cos(3)(
)77cos()(
2
1
ttx
ttx
Copyright Monash University 2009
POP QUIZ (answer)
COMPLEXADDITION:
CONVERTbacktocosineform:
16
j 3 3e j 0.51
31 j 3/231 jej
)77cos(2)( 33 ttx
Copyright Monash University 2009
ADD SINUSOIDS EXAMPLE
17
tm1
tm2
tm3
)()()( 213 txtxtx
)(1 tx
)(2 tx
Copyright Monash University 2009
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
18
Copyright Monash University 2009
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
19
Copyright Monash University 2009
ADD SINUSOIDS
20
VECTOR(PHASOR)ADD
X1
X2
X3