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V/R)e
-
a
t 
+ 
V/R 
= 
I
0
e
-
a
t
+ 
V/R (1 
- 
e
-
a
t
) for t 
≥
0
+
.
Thus, the 
correct solution is, 
i t
I e
V
R
e
t
t
t
L
for
( )
(
)
/
/
=
+
−
≥
−
−
+
0
1
0
t
t
(10.6-1)
The first term in Eqn. 10.3-8 does not get multiplied by V when the DC voltage magnitude is scaled 
by V. The second term in the same equation gets scaled by V. This means that the forced response (the 
constant term in the solution) obeys the principle of homogeneity.
Now, let us solve the circuit for three situations (a) with V
1 
(b) V
2 
and (c) V
1 
+ 
V
2 
as the DC voltages 
with the same initial condition for all the three cases. The expression for i
L
(t) in the three cases can 
be derived as 
Case (a)
for
Case (b)
L
i t
I e
V
R
e
t
i
t
t
( )
(
)
/
/
=
+
−
≥
−
−
+
0
1
1
0
t
t
L
L
L
for
Case (c)
( )
(
)
( )
/
/
/
t
I e
V
R
e
t
i t
I e
t
t
t
=
+
−
≥
=
−
−
+
−
0
2
0
1
0
t
t
t
++
+
−
≥
−
+
V V
R
e
t
t
1
2
1
0
(
)
/
t
for
(10.6-2)
These expressions show that the forced response part obeys the principle of additivity also. In all 
cases we see that neither the total response nor the transient response (or natural response) obeys 
superposition principle.
We have arranged the terms in the expression for inductor current in Eqn. 10.6-1 and Eqn. 10.6-2
in a special manner – the dependence on initial condition is contained in the first term and the 
dependence on step magnitude is contained in the second term. We notice that it is not only the forced 

---

Linearity and Superposition Principle in Dynamic Circuits 
10.33
response component which satisfies the superposition principle, but the entire second term which 
depends on forcing function satisfies the superposition principle. However, both terms contribute to 
transient response and transient response does not satisfy superposition principle. 
We notice further that the first term depends only on initial condition and will be the total response 
if there is no forcing function, i.e., the first term is the response in a source-free circuit. Similarly, the 
second term depends on forcing function and does not depend on initial condition. The second term 
will be the total response if the circuit is initially relaxed. These observations will remain valid for 
any forcing function. The nature of the second term will change with the nature of forcing function. 
However, the resolution of total response into two components –one that depends entirely on initial 
condition and another that depends entirely on forcing function – will be possible for any forcing 
function in any linear circuit. 
Here too we accept a result proved in the general theory of linear systems without worrying about 
its proof. 
The response for 
t
≥
0
+ 
that results from initial condition alone (that is, with zero input 
for 
t
≥
0
+
) is called ‘
zero-input response
’. The response for 
t
≥
0
+ 
that results from 
application of input for 
t
≥
0
+ 
with zero initial condition is called ‘
zero-state response
’. 
The total response in a linear time-invariant circuit containing energy storage elements 
can be found by adding the zero-input response and zero-state response together. 
Zero-input response will depend only on the initial state of the circuit as encoded in its 
initial condition specifications. Zero-state response will depend only on forcing function.
Now, we focus on the zero-input response of an RL circuit. This is the response in a source-free 
circuit due to its initial energy alone. It is I
0
e
-
a
t
A with the usual interpretations for all the symbols. 
It must be obvious that the zero-input response will scale with I
0
, i.e., when the initial condition 
value is multiplied by a real constant the zero-input response also gets multiplied by the same 
constant. Similarly, when two different values of initial condition I
01
and I
02
result in two different 
zero-input responses, the zero-input response with the initial condition value at I
01
+ 
I
02
will be the 
sum of the two zero-input responses observed in the first two cases. Thus, zero-input response of RL 
circuit (and, of all linear time-invariant circuits) obeys superposition principle with respect to initial
condition values.
Thus, we see that both zero-state response and zero-input response obey superposition principle 
individually.
Zero-input response follows superposition principle with respect to initial condition 
values and zero-state response obeys superposition principle with respect to input 
source functions. 
Therefore, total response will not follow superposition principle with respect to forcing function or 
initial condition – only its components will obey superposition principle. 
Figure 10.6-1 shows the zero-input response and zero-state response components along with 
the total current response in an RL circuit under DC voltage switching condition for various 
values of normalised initial condition values. Decomposition of total response into transient 
response and forced response for the same circuit was shown in Fig. 10.4-4. Compare these two
decompositions. 

---

10.34
First-Order 
RL
Circuits
–0.5
(–0.5)
(–0.5)
1
(0.5)
Zero-input response
Zero-state response
Total response
(0.5)
(1.5)
(1.5)
i
Ln
t
/
τ
–0.25
0.25
0.5
0.75
1
1.25
1.5
2
3
Fig. 10.6-1 
Decomposition of total response into zero-input response and zero-state 
response
Both zero-input response and zero-state response will contain natural response terms. 
However, the natural response component in zero-state response has amplitude that 
depends on forcing function value and 
does not 
depend on initial condition value. Zero-
input response and part of zero-state response together will form transient response. 
The remaining part of zero-state response will be the forced response.