0195136047.pdf

5.4 APPLICATIONS OF OPERATIONAL AMPLIFIERS 253

where

fl=

1

2 πRfCf

(5.4.43)

The magnitude
∣
∣
∣
∣

vo

vi

∣

∣

∣

∣=

Rf

Ri

1

√

1 +(f/fl)^2

is sketched in Figure 5.4.14, from which one can see that at the half-power or 3-dB point offl,
the dc valueRf/Rigoes down by a factor of 1/

√
2 or 0.707.
WithZf=RfandZi=Ri+ 1 /j ω Ci(i.e.,RiandCiin series), we have ahigh-pass filter
for which we obtain
vo
vi

=−

Rf

Ri

jf/fh

1 +(jf/fh)

(5.4.44)

where

fh=

1

2 πRiCi

(5.4.45)

and
∣
∣
∣
∣

vo

vi

∣

∣

∣

∣=

Rf

Ri

1

√

1 +(fh/f)^2

(5.4.46)

which is sketched in Figure 5.4.15. Note that the high-frequency gain isRf/Ri, as in the low-filter
case, and the half-power or 3-dB point isfh.

vo

vi

Rf

Ri

Rf

0.707R

i

f

fl

Figure 5.4.14Frequency response of voltage-gain magni-

tude of a low-pass active filter.

vo

vi

Rf

Ri

Rf

0.707R

i

f

fh

Figure 5.4.15Frequency response of voltage-gain

magnitude of a high-pass active filter.