0195136047.pdf

3.2 TRANSIENTS IN CIRCUITS 141

f(t)

t

A

0 T

Figure 3.2.6Rectangular pulse.

The student is encouraged to justify this statement by drawing graphically. For determining the
response of a circuit to a pulse excitation, using the principle of superposition, the response to each
component can be found and the circuit response obtained by summing the component responses.
The decomposition of pulse-type waveforms into a number of step functions is commonly used.
In order to represent the effects of pulses of short duration and the responses they produce,
the concept ofunit-impulsefunctionδ(t) is introduced,

δ(t)=0 fort= 0 , and

∫∞

−∞

δ(t)dt=

∫ 0 +

0 −

δ(t)dt= 1 (3.2.44)

Equation (3.2.44) indicates that the function is zero everywhere except att=0, and the area
enclosed is unity. In order to satisfy this,δ(t) becomes infinite att=0. The graphical illustration
of a unit-impulse function is shown in Figure 3.2.7. The interpretation ofi(t)=Aδ(t−T)is
thatδ(t−T)is zero everywhere except att=T(i.e., the current impulse occurs att=T), and
the area underi(t)isA.

f(t)

t

1

1

− 2 ∆

(^01)
2
(a)
∆
u(t)
t
1
0
(b)
As ∆ → 0
df(t)
dt
t
1
− 2 ∆
(^01)
2
(c)
∆
1
∆
δ(t) = du(t)
dt
t
0
(d)
As ∆ → 0
Figure 3.2.7Graphic illustration of unit-impulse function.(a)A modified unit-step function in which the
transition from zero to unity is linear over a -second time interval.(b)Unit-step function: the graph of
part (a) as →0.(c)The derivative of the modified unit-step function depicted in part (a) (area enclosed
=1).(d)Unit-impulse function: the graph of part (c) as →0[
∫∞
−∞δ(t)dt=
∫ 0 +
0 −δ(t)dt=1]