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7.2.1 
sinusoidal steady-state response from response to 
e 
j
v
 t
These two ways of expressing trigonometric functions in terms of complex exponential functions 
suggest two methods to obtain the sinusoidal steady-state response in dynamic circuits. The first 
method is to obtain steady-state response to complex exponential inputs e 
j
w
 t
and e
-
j
w
 t
and obtain the 
steady-state response for cos
w
 t as the sum of responses for e 
j
w
 t
and e
-
j
w
 t
. But this will be correct if and 
only if the particular integral of a linear constant-coefficient differential equation obeys superposition 
principle. The mathematical theory of such differential equations assures us that it is indeed so.

---

7.8
The Sinusoidal Steady-State Response
The 
steady-state 
response 
component 
in 
linear 
time-invariant 
circuits 
obeys 
superposition principle.
The second method will be to obtain the steady-state response for a cos
w
 t input as the real part 
of steady-state response to a complex exponential function e 
j
w
t
and the steady-state response for a 
sin
w
 t input as its imaginary part. The underlying reasoning is that since cos
w
 t is the real part of 
e 
j
w
t
, the response for cos
w
 t must be the real part of response for e 
j
w
t
. It looks intuitively evident. 
But this turns out to be true only for linear circuits. This will be true only if the real part of the input 
function does not affect the imaginary part of response and the imaginary part of input function does 
not affect the real part of response. We employ the superposition principle for particular integral of 
a constant-coefficient linear differential equation to verify that it is so in the case of such differential
equations.
e 
j
w
t
can be viewed as a linear combination of two input functions 
-
cos
w
 t multiplied by 1 added 
to sin
w
 t multiplied by j. We can view the steady-state response of a linear circuit to e 
j
w
t
input as 
the particular solution of the describing differential equation of the circuit to a composite input – 
cos
w
 t multiplied by 1 added to sin
w
 t multiplied by j. But, particular solution obeys superposition 
principle. Therefore, steady-state response to e 
j
w
t
=
steady-state response to cos
w
 t 
+ 
j times steady-
state response to sin
w
 t. Therefore,
The 
real part
of steady-state response to 
e 
j
w
t
=
steady-state response to cos
w
 t
and 
the 
imaginary part
of steady-state response to 
e 
j
w
t
=
steady-state response to sin
w
 t
.
Therefore, the second method for determining the sinusoidal steady-state response in terms of 
steady-state response to complex exponential function will yield correct result for linear circuits.