Signals and Systems - Electrical Engineering

2.3 LTI Continuous-Time Systems 151

If we then let 1 →0 we have that (recall thatds(t)/dt=h(t)is the relation between the unit-step

responses(t)and the impulse responseh(t))

lim

1 → 0

y(t)=

ds(t)

dt

=h(t)=

1

T

[u(t)−u(t−T)]

That is, this system approximates the impulse response of the averager. n

nExample 2.15

Consider the circuits obtained with an operational amplifier when we feed back its output with a

wire, a resistor, and a capacitor (Figure 2.16). Assume the linear model for the op-amp. The circuits

in Figure 2.16 are called avoltage follower, anintegrator, and anadder.

Solution

Virtual follower circuit.Although the operational amplifier can be made linear, its large gain

Amakes it not useful. Feedback is needed to make the op-amp useful. Thevoltage followercircuit

(Figure 2.16(a)), which is used to isolate cascaded circuits, is a good example of a feedback system.

Given that the voltage differential is assumed to be zero, thenv−(t)=vi(t), and therefore the

output voltage is

vo(t)=vi(t)

FIGURE 2.16

Operational amplifier circuits:

(a) virtual follower, (b) inverting

integrator, and (c) adder with

inversion.

+

+

−

−

−

−

+

−

vi(t)

vo(t)=vi(t)

(a)

R

C

+ +

−

vi(t)

+

−

vo(t)

(b)

(c)

R

+

+

− +

−

+

−

vo(t)

v 1 (t)

v 2 (t)

R 1

R 2