California State University, Bakersfield
Signals and Systems
Vida Vakilian
Department of Electrical and Computer Engineering,
California State University, Bakersfield
Lecture 3 (Phasors)
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California State University, Bakersfield
Signals and Systems
Complex Numbers
We will find it is useful to represent
sinusoids as complex numbers
j = −1
z = x + jy
Rectangular coordinates
z = z ∠θ = z e jθ
Polar coordinates
Re(z ) = x
Im( z ) = y
Relations based on
Euler’s Identity
e ± jθ = cosθ ± j sin θ
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California State University, Bakersfield
Signals and Systems
Complex Numbers
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California State University, Bakersfield
Signals and Systems
Complex Numbers
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California State University, Bakersfield
Signals and Systems
Complex Numbers
Learn how to perform these
with your calculator/
computer
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California State University, Bakersfield
Signals and Systems
Complex Numbers
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California State University, Bakersfield
Signals and Systems
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California State University, Bakersfield
Signals and Systems
Outline
Ø Phasors
Ø RLC
circuit
Ø Traveling
waves in phasor domain
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California State University, Bakersfield
Signals and Systems
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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California State University, Bakersfield
Signals and Systems
Phasor Domain
Ø  The phasor technique can also be used for analyzing linear
systems when the forcing function is an arbitrary (nonsinusoidal) 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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California State University, Bakersfield
Signals and Systems
Phasor Domain
Phasor counterpart of
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California State University, Bakersfield
Signals and Systems
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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California State University, Bakersfield
Signals and Systems
Phasor Relation for Resistors
Current through resistor
Time Domain
Phasor Domain
Time domain
i = I m cos (ω t + φ )
υ = iR = RI m cos (ω t + φ )
Phasor Domain
V = RI m∠φ
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California State University, Bakersfield
Signals and Systems
Phasor Relation for Inductors
Time domain
Phasor Domain
Time Domain
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California State University, Bakersfield
Signals and Systems
Phasor Relation for Capacitors
Time domain
Time Domain
Phasor Domain
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California State University, Bakersfield
Signals and Systems
AC Phasor Analysis: General Proc.
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California State University, Bakersfield
Signals and Systems
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California State University, Bakersfield
Signals and Systems
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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California State University, Bakersfield
Signals and Systems
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