Page 125 - Advanced engineering mathematics

3.5 Impulses and the Delta Function 105

Transients can be generated in a circuit during switching and can be harmful because they
contain a broad spectrum of frequencies. If one of these is near the natural frequency of the
system, introducing the transient can cause resonance to occur, resulting in oscillations large
enough to cause damage. For this reason, engineers sometimes use a delta function to model a
transient and study its effect on a circuit being designed.

EXAMPLE 3.18
Suppose the current and charge on the capacitor in the circuit of Figure 3.24 are zero at time
zero. We want to describe the output voltage response to a transient modeled by δ(t). The output
voltage is q(t)/C, so we will determine q(t). By Kirchhoff’s voltage law,
1

Li + Ri + q = i + 10i + 100q = δ(t).
C

Since i = q, then

q + 10q + 100q = δ(t).
Assume the initial conditions q(0) = q (0) = 0. Apply the transform to the initial value problems

to get
2
s Q(s) + 10sQ(s) + 100Q(s) = 1.
Then
1 1
Q(s) = = .
2
s + 10s + 100 (s + 5) + 75
2
The last expression is preparation for shifting in the s− variable. Since
√
1 1
−1
L = √ sin(5 3t),
s + 75 5 3
2
then
√
1 1
−1 −5t
q(t) = L = √ e sin(5 3t).
2
(s + 5) + 75 5 3
The output voltage is
1 20 √
q(t) = 100q(t) = √ e −5t sin(5 3t).
C 3
A graph of this output voltage is given in Figure 3.25. The circuit output displays damped
oscillations at its natural frequency even though it was not explicitly forced by oscillation of this
frequency.

1 H 10 Ω

0.01 F

FIGURE 3.24 Circuit of Example 3.18 with
E (t) = δ(t).
in

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October 14, 2010 14:14 THM/NEIL Page-105 27410_03_ch03_p77-120