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t
1
Since this equation is true for all t (it is a KVL equation), we can integrate both sides from 
-∞
to 
t and write
di t
dt
dt
i t dt
L
v
t dt
t
t
t
SU
t
L
L
for all 
( )
( )
( )




+
=
−∞
−∞
−∞
∫
∫
∫
1
1
t
Changing the order of integration and differentiation
d
dt
i t dt
i t dt
L
v
t dt
t
t
t
SU
t
L
L
for all 
( )
( )
( )
−∞
−∞
−∞
∫
∫
∫
+
=
1
1
t
Let
and
then
L
L
ˆ ( )
( )
ˆ ( )
( ) ,
i t
i t dt
v
t
v
t dt
t
SU
SU
t
=
=
−∞
−∞
∫
∫
di t
dt
i t
L
v
t
t
SU
ˆ ( ) ˆ ( )
ˆ ( )
L
L
for all 
+
=
t
1
The last equation is the differential equation describing the zero-state response for integrated input 
function. 
Therefore, we can state that the zero-state response in a linear circuit for an integrated 
input function is equal to integral of the zero-state response for the input function. 

---

Unit Impulse Response of Series 
RL
Circuit 
10.39
Strictly speaking, we showed this only for a first-order differential equation; but there is nothing 
in the proof which limits it to first-order differential equation alone. This result is valid for any 
linear lumped circuit described by linear differential equation of any order. The integration has to be 
performed from 
-∞
in theory. However, since we know that the zero-state response is zero from 
-∞
to 
t 
= 
0
-
, we need to integrate from t 
= 
0
-
to t only.
Integral of 
d
(t) gives u(t). Therefore, integral of impulse response should give unit step response 
(usually referred to as the step response). This is verified by carrying out the integration below.
Zero-state unit impulse response of an RL circuit 
=
−
1
L
e
t
/
t
ffor
and for
Integrating for a 
,
t
t
t
 dt
a
≥
≤
>
+
+
−
+
∞
∫
0
0
0
0
0
0
−
−
(
bbounded number dt
L
e
dt
L
e
R
e
t
t
t
t
)
(
)
(
/
/
/
+
=
−
=
−
−
−
−
+
+
∫
∫
1
1
1
1
0
0
0
t
t
t
−
tt
) for 
t
≥
+
0
Unit ramp input function is defined as 
r t
t
t
t
( )
=
<
≥



0
0
0
for
for
.
It can be easily verified that unit ramp function is the integral of unit step function. These three 
basic input functions and their relations are shown in Fig. 10.7-2. 
t
(
t
)
δ
1
u
(
t
)
1
d
t
d
t
t
d
t
d
(
t
)
r
1
1
Fig. 10.7-2 
Impulse, step and ramp functions and their relations
Therefore, the ramp response can be found by integrating the step response as below.
Ramp response
for
=
−
=
−
−


≥
+
∫
−
−
+
1
1
1
1
0
0
R
e
dt
R
t
e
t
t
t
t
(
)
(
)
/
/
t
t
t
Ramp response and its components are shown in Fig. 10.7-3. The voltage across resistor is plotted 
instead of inductor current.
The unit ramp function has a kink at t 
= 
0 and hence it is not differentiable at t 
= 
0. However, 
it is differentiable at all other time instants. Hence, first 
derivative of r(t) will be a function defined as 0 for t 
≤
0
-
,
1 for t 
≥
0
+ 
and undefined at t 
= 
0. But that is the unit step 
function. Therefore, unit step function is the first derivative 
of unit ramp function.
Unit step function is not even continuous at t 
= 
0. 
Obviously, it cannot be differentiated there. However, we 
raise the question – which function when integrated will 
yield unit step function? The answer is that it is the impulse 
function. Therefore, we can consider 
d
(t) to be the first 
derivative of u(t) in the ‘anti-derivative’ sense.
Fig. 10.7-3 
Unit ramp response 
of 
RL
circuit
v
R
(
t
)
t
t
t – 
(1 – e )
 
τ
–
τ
(1 – e )
 
–t
τ
τ
t

---

10.40
First-Order 
RL
Circuits
KVL equations are true for all t. Both sides of an equation which holds for all t can be differentiated 
with respect to time. From this observation, it is easy to see that the following is true.
The zero-state response in a linear circuit for differentiated input function is equal to 
the derivative of the zero-state response for the input function.
Therefore, we can get to unit step response and unit impulse response in any linear circuit by 
successive differentiation of its unit ramp response.