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7.4.2 
element relations in terms of complex amplitudes
Consider a resistor. Passive sign convention is assumed everywhere. The element equation of a resistor 
is v
R
=
Ri
R
. Substituting 
v
V e
e
i
I e
e
j
j t
j
j t
v
i
R
Rm
R
Rm
and 
=
=
(
)
(
)
f
w
f
w
we get 
(
)
(
)
V e
e
R I e
e
j
j t
j
j t
v
i
Rm
Rm
f
w
f
w
=
and hence V
RI
v
i
Rm
Rm
and 
=
=
f f
.
This implies that a resistor can not bring about a phase difference 
between the current and voltage variables. Using the bold face italic notation for the complex 
amplitude, we write the element relation of a resistor for complex amplitudes as follows:
V
I
R
R
=
R  
(7.4-1)
The element equation of an inductor is 
v
L
di
dt
L
L
=
.
Substituting 
v
V e
e
i
j
j t
v
L
Lm
L
and 
=
=
(
)
f
w
(
)
I e
e
j
j t
i
Lm
f
w
we get 
(
)
(
)(
)
V e
e
L j
I e
e
j
j t
j
j t
v
i
Lm
Lm
f
w
f
w
w
=
. This implies that V
LI
Lm
Lm
v
=
=
w
f
and 
f
i
+ °
90 .
Thus an inductor scales the current amplitude by 
w
L and adds a 90
°
phase advance to the current to 
generate the voltage across it. Using the bold face italic notation for the complex amplitude, we write 
the element relation of an inductor for complex amplitudes as below. 
V
I
I
I
V
V
L
L
L
L
L
L
e
e
=
=
=
=






°
−
°
(
)
(
)
j L
L
j L
L
j
j
w
w
w
w
90
90
1
1
(7.4-2)
Thus, the inductor operates upon the complex amplitude of current by the operator j
w
L to generate 
the voltage complex amplitude across itself.
The amplitude of current in an inductor is 1/
w
L
times the amplitude of voltage and the 
current 
lags
the voltage by 90
°
under sinusoidal steady-state condition.
The element equation of a capacitor is 
i
C
dv
dt
C
C
=
.
Substituting 
v
V e
e
i
j
j t
v
C
Cm
C
and 
=
=
(
)
f
w
(
)
I e
e
j
j t
i
Cm
f
w
we get 
(
)
(
)(
)
I e
e
C j
V e
e
j i
j t
j
j t
v
Cm
Cm
f
w
f
w
w
=
. This implies that I
CV
i
v
Cm
Cm
and 
=
= + °
w
f f
90 .
Thus a capacitor scales the voltage amplitude by 
w
C and adds a 90
°
phase advance to the voltage to 
generate the current through itself. Using the bold face italic notation for the complex amplitude, we 
write the element relation of a capacitor for complex amplitudes as below.

---

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.