0195136047.pdf

252 ANALOG BUILDING BLOCKS AND OPERATIONAL AMPLIFIERS

vo(t)=−

1

C

∫t

−∞

iC(ξ)dξ=−

1

RC

∫t

−∞

vin(ξ)dξ (5.4.39)

which illustrates that the network behaves as an integrator with sign inversion.

Differentiator

Shown in Figure 5.4.12 is a differentiator which is obtained by replacingR 1 in the inverting

amplifier of Figure 5.4.1 by a capacitorC. Assuming ideal op-amp characteristics, one hasi=iC

andvo=−Ri=−RiC. But sinceiC(t)=Cdvin(t)/dt,weget

vo(t)=−RC

dvin(t)

dt

(5.4.40)

which corresponds to a differentiator with a gain of−RC. In practice, however, differen-

tiators are normally avoided because of high-frequency noise (which is accentuated due to

a transfer function that increases with frequency) and stability problems (which make them

oscillate).

Inductorless (Active) Filters

Filters (used to pass or eliminate certain frequency components of a signal) that are suitable

for IC fabrication, but which do not contain inductors, are known asactive filters. They have

the op amp as a common component. Figure 5.4.13 shows the basic op-amp circuit with

frequency-dependent impedances. The voltage gain or voltage transfer function of this circuit

is given by

vo

vi

=−

Zf

Zi

(5.4.41)

withZi =RiandZf =Rf‖( 1 /j ω Cf)(i.e.,Rf andCf in parallel). We have alow-pass

filterwith

vo

vi

=−

Rf

Ri

1

1 +jωRfCf

=−

Rf

Ri

1

1 +(jf/fl)

(5.4.42)

−

+

vo

1

2 3

C

i i

C

R

vin

Figure 5.4.12Differentiator.

−

+

+

vo

+

−

vi

Zf

Zi

−

Figure 5.4.13Basic op-amp active filter.