Vida Vakilian - California State University, Bakersfield · Lecture 3 (Phasors) Signals and Systems...
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Signals and Systems 1 California State University, Bakersfield Vida Vakilian Department of Electrical and Computer Engineering, California State University, Bakersfield Lecture 3 (Phasors)
Transcript of Vida Vakilian - California State University, Bakersfield · Lecture 3 (Phasors) Signals and Systems...
Signals and Systems
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California State University, Bakersfield
Vida Vakilian
Department of Electrical and Computer Engineering, California State University, Bakersfield
Lecture 3 (Phasors)
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California State University, Bakersfield
Complex Numbers We will find it is useful to represent sinusoids as complex numbers
jyxz +=θθ jezzz =∠=
1−=j
Rectangular coordinates Polar coordinates
θθθ sincos je j ±=±
Relations based on Euler’s Identity
( )yzxz
=
=
)Im(Re
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Complex Numbers
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California State University, Bakersfield
Complex Numbers
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California State University, Bakersfield
Complex Numbers
Learn how to perform these with your calculator/computer
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Complex Numbers
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Outline
Ø Phasors
Ø RLC circuit
Ø Traveling waves in phasor domain
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Phasor Domain
Ø The phasor-analysis technique transforms equations from the time domain to the phasor domain.
Ø Integro-differential equations get converted into linear equations with no sinusoidal functions.
Ø After solving for the desired variable--such as a particular voltage or current-- in the phasor domain, conversion back to the time domain provides the same solution that would have been obtained had the original integro-differential equations been solved entirely in the time domain.
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Phasor Domain
Ø The phasor technique can also be used for analyzing linear systems when the forcing function is an arbitrary (non-sinusoidal) periodic time function.
Ø By expanding the forcing function into a Fourier series of sinusoidal components we can solve for the desired variable using phasor analysis and superposition.
Ø Moreover, for non-periodic source functions, such as a single pulse, the functions can be expressed as Fourier integrals.
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Phasor Domain
Phasor counterpart of
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Time & Phasor Domain It is much easier to deal with exponentials in the phasor domain than sinusoidal relations in the time domain Just need to track magnitude/phase, knowing that everything is at frequency w
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Phasor Relation for Resistors
Time Domain Phasor Domain ( )φωυ +== tRIiR cosm
φ∠= mRIV
Current through resistor
( )φω += tIi cosm
Time domain
Phasor Domain
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Phasor Relation for Inductors
Time Domain
Time domain
Phasor Domain
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Phasor Relation for Capacitors
Time Domain
Time domain
Phasor Domain
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AC Phasor Analysis: General Proc.
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Traveling Waves
Ø We know the left hand side expresses a wave moving in the negative x direction.
Ø In the phasor domain a wave of amplitude A traveling in a lossless domain moving in the positive x direction is given by and a wave moving in the neg x direction is represented by . Thus the sign of x in the exponent is opposite to the direction of travel.
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