PHASOR CONCEPT
FOR
AC STEADY-STATE ANALYSIS
PROF. SIRIPONG POTISUK
ELEC 308
First-order LCCDE’s
y Standard form:
dx (t )
+a
ax (t ) = f (t )
d
dt
x (t ) – unknown variable of interest
y f (t ) – forcing or excitation function
y a – constant coefficient depending upon the
y
makeup of the system
y Solution:
x (t ) = ko e −at + e −at ∫ e aτ f (τ )dτ
123
homogeneous
14t42443
particular
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Sinusoidal Steady-state response
y A steady-state response of a linear circuits has the
same functional form as the forcing function
y A sinusoidal forcing function Vm cos (ω t + θ)
produces a steady-state response Im cos (ω t + φ)
y Only the amplitude and phase are appropriately
scaled; the frequency does not change
Phasors
y Provides a simple means in obtaining the steady-state
espo se o
of linear
ea ccircuits
cu ts excited
e c ted by ssinusoidal
uso da
response
sources
y A complex representation of a sinusoidal function
expressed using the cosine standard
y A complex number in polar form that is used to
compactly represent the amplitude and phase of a
sinusoid
d simultaneously
l
l
y The angular frequency of the sinusoid is suppressed,
yet implicitly present
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The Transform Solution Process
Phasor Transformation
y The process of representing a sinusoidal function in
t e time
the
t e domain
do a as a p
phasor
aso in tthee p
phasor
aso o
or
frequency domain
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Example 6.7
Determine the corresponding phasor for
a. v(t)
(t) = 150cos(120πt+30°)
(
t
°)
b. v(t) = 312 sin(120πt+45°)
Example 6.8
Given v1(t) = 5cos(ωt+45°) and v2(t) = 10cos(ωt+30°),
calculate v(t)
v(t)= v1(t)
(t)+ v2(t).
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Phase Relationships Via Phasors
y Can be determined from the so-called ‘phasor
diagram’
diagram
y Consider the phasors to rotate counterclockwise
when standing at a fixed point
y If V1 arrives first followed by V2 after a rotation of
and angle of θ, we say that V1 leads V2 by θ
y Alternatively,
Alternatively we could say that V2 lags V1 by θ
y Usually, θ is taken as the smaller angle between
the two phasors
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Phasor Relationships for Circuit Elements
y In analyzing AC circuits using the phasor concept
to obtain its steady
steady-state
state response
response, the entire
circuit model needs to be transformed into the
phasor domain
y How does phasor transformation affect the v-i
characteristics of passive circuit elements?
y The phasor transformation of a passive element
results in a complex number called complex
impedance in.
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Complex Impedances
y The impedance of a circuit element is the ratio of the
y
y
y
y
phasor voltage across to the phasor current through
the element (Ohm’s law in the phasor domain)
A resistance-like quantity, i.e., behaves like
resistance in DC circuits
Denoted by Z and measured in ohms (Ω)
Represents the ability of the element to oppose the
flow of sinusoidal current
Impedance is not a phasor because it cannot be
inverse-transformed into a sinusoidal function
Z X = R + jX
Reactance
Impedance
Resistance
YX = G + jB
Admittance
Susceptance
Conductance
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Complex Impedance for Resistor
y The v-i characteristics of a resistor in the phasor
domain continues to obey Ohm’s
Ohm s law as in the time
domain
y A resistor R is represented by an impedance of the
same value, i.e., ZR = R
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Complex Impedance for Inductor
y The v-i characteristics of an inductor in the phasor
domain shows a phase shift of 90
90° between the
voltage across and the current through it (the term j)
while the magnitude is scaled by ωL
y The impedance of an inductor L is ZL = jωL
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Complex Impedance for Capacitor
y The v-i characteristics of an inductor in the phasor
domain shows a phase shift of 90
90° between the
voltage across and the current through it (the term j)
while the magnitude is scaled by 1/ωC
y The impedance of an inductor L is ZC = 1/jωC
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Example 6.9
A voltage v(t) = 110cos(120t+45°) is applied to a 47-μF
capacitor. Find the impedance of the capacitor.
Series Impedance Combination
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Parallel Impedance Combination
Example 6.10
A voltage v(t) = 220cos(360t+30°) is applied to the
circuit shown. Find the total impedance of the series
inductance and resistance.
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Example 6.11
A voltage v(t) = 10cos(500t+60°) is applied to the circuit
shown. Find the total impedance as seen by the source.
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Phasor Transform Solution Process
y Transform the time-domain voltage and current
sources into phasors (All sources must have the same
frequency and be expressed using cosine standard)
y Replace all passive elements with their impedances
y Analyze the resulting phasor-domain circuit using
any of the DC analysis techniques studied earlier
((KVL,, KCL,, Ohm’s,, voltage
g and current division,,
network reduction, etc.), but performing the
calculations with complex arithmetic
y Inverse-transform the solution back to time domain
Example 6.13
Find the steady-state current i(t) and voltage vL(t).
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Example 6.15
Find the steady-state current through the inductor and
the capacitor and voltage across the capacitor, vC(t).
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