7.14
The Sinusoidal Steady-State Response
I
V
V
V
I
I
C
C
C
C
C
C
=
=
=
=
°
−
°
(
)
(
)
j C
C
j C
C
j
j
w
w
w
w
e
e
90
90
1
1
(7.4-3)
Thus, the capacitor
operates upon the
complex amplitude of voltage by the
operator j
w
C to generate
the complex amplitude of current through it.
The amplitude of current in a capacitor is
w
C
times the amplitude
of voltage and the
current
leads
the voltage by 90
0
under sinusoidal steady-state condition.
7.4.3
the phasor
Electrical Engineers decided long back that a new name is required for what we have been calling the
complex amplitude. Complex amplitude is the number that gives the amplitude of complex exponential
function and the phase of the complex exponential function in the form of a single complex number.
Its magnitude gives the amplitude of the signal and its angle gives the phase of the signal. The phase
is referred to the standard
e
j
w
t
reference function. Electrical Engineers call complex amplitude
a
phasor.
Phasor
is a complex number that gives the amplitude of complex exponential function
and the phase of the complex exponential function with
the time-variation of the
function understood as
e
j
w
t
. It can be used as a representation for a sinusoidal
function.
Thus
phasor is just a new name for what we have understood till now as
complex amplitude.
The process of starting with a sinusoidal function and ending up with its
phasor representation is
called
Phasor Transformation. We summarize the steps involved in this transformation.
• Express the given sinusoidal function in the form
x
(
t
)
=
X
m
cos (
w
t
+
q
).
• Write
x
(
t
) as the
real part
of
X
m
e
j
(
w
t
+
q
)
.
• Suppress the qualifier
real part
.
• Suppress
e
j
w
t
after noting the value of
w
for later use.
•
The resulting complex number
X
=
X
m
e
j
q
is the
phasor representation
for
x
(
t
). The bold
face italic notation stands for the magnitude and angle together. The symbol
X
may
be used in hand-written text.
Once we get the answer for a circuit analysis problem in
phasor representation, we need to go back
to time-domain to get the time-domain output that we wanted really. We start in the time-domain, we
want to end up in the time-domain and the netherworld of
phasors is only a temporary sojourn. The
steps
involved in inverse phasor transformation are listed below.
Transforming a Circuit into Phasor Equivalent Circuit
7.15
• Obtain the magnitude
X
m
and angle
q
of the
phasor
and put it in
X
m
e
j
q
form.
• Multiply
X
m
e
j
q
by
e
j
w
t
and express it in
X
m
e
j
(
w
t
+
q
)
form.
• Get the
real part
as
X
m
cos (
w
t
+
q
) by using Euler’s Identity.
We are free to express the time-function as
x(
t)
=
X
m
sin(
w
t
+
q
). The
phasor representation will
remain the same. But
imaginary part will be implied everywhere.
The steps in forward and inverse
phasor transformation are easy and can be done by inspection
with a little practice. For instance, if
X
=
1
+
j1, then the magnitude is
√
2 and angle is 45
°
and the time-
domain waveform is
√
2cos(
w
t
+
45
°
) if we started with a cosine and it is
√
2sin(
w
t
+
45
°
) if we started
with a sine. Of course we need to know the value of
w
in addition to
X
. The phasor representation
of
a sinusoidal waveform will not contain the frequency information. Frequency has to be known
separately.
7.5
transFormInG a cIrcuIt Into phasor equIvalent cIrcuIt
We have already seen that we can write the KVL and KCL equations directly in terms of
complex
amplitudes (
i.e., phasors) and that there are well-defined relations between
complex voltage amplitude
(
i.e., voltage phasor) and
complex current amplitude (
i.e., current phasor) for all two-terminal elements.
The ratio of
voltage phasor to
current phasor is equal to
R in the case of a resistor. It is
j
w
L in the
case of an inductor and it is 1/
j
w
C in the case of a capacitor.
These facts suggest that we need not write down the mesh and node equations in time-domain at
all. We can write them in terms of mesh current phasors or node voltage phasors using the element
relation that ties up voltage phasor of the element to current phasor of the element. The resulting
equations will be algebraic equations involving phasors. Thus
phasor transformation of all circuit
variables results in a set of simultaneous algebraic equations rather than simultaneous differential
equations. Eliminating variables and solving for the desired circuit variable is far easier when we deal
with algebraic equations than when we deal with differential equations.
The circuit that will help us to deal with steady-state response to complex exponential
input as if it is a memoryless circuit is called the
phasor equivalent
circuit
of the dynamic
circuit.
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