0195136047.pdf

254 ANALOG BUILDING BLOCKS AND OPERATIONAL AMPLIFIERS

vo

f 0

High pass

Resultant

bandpass

characteristic

Low pass

vi

f

Figure 5.4.16Bandpass filter seen as a combina-

tion of low-pass and high-pass filter characteris-

tics.

By using a low-pass and a high-pass filter and overlapping their transfer functions, as shown

in Figure 5.4.16, one can construct abandpass filterwith

Zf=Rf

∥

∥

∥

∥

1

jωCf

=

Rf

1 +jωRfCf

(i.e.,RfandCfin parallel)

and

Zi=Ri+

1

jωCi

(i.e.,RiandCiin series)

in which case

vo

vi

=−

Zf

Zi

=−

Rf

Ri

1

(

1 +jωRfCf

)

[ 1 + 1 /(jωRiCi)]

(5.4.47)

Denoting

fl=

1

2 πRiCi

(5.4.48a)

and

fh=

1

2 πRfCf

(5.4.48b)

Equation (5.4.47) can be rewritten as

vo

vi

=−

Rf

Ri

1

( 1 +fl/fh)+j(f/fh−fl/f)

=−

Rf

Ri

fh/(fh+fl)

1 +j(f^2 −flfh)/[f(fl+fh)]

(5.4.49)

whose magnitude is sketched in Figure 5.4.17. Notice that the magnitude response is a maximum

with a value ofRf/Riatf 0 =

√

flfh, and iffl<< fh, then the magnitude becomes( 0. 707 Rf/Ri

atflandfh. The bandwidth (BW) may then be defined as