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A
Band-Pass Filter
is a circuit that only allows a certain range or band of frequencies
to pass, while attenuating or rejecting signals whose frequencies
are either below a lower cut-off frequency or above an upper cut-off
frequency. A simple band-pass filter may be obtained by combining a
low-pass filter and a
high-pass filter. The range of
frequencies that a band-pass filter allows to pass is referred to as
the
'passband'.
The band-pass filter is the opposite of the
band-stop filter.
An ideal band-pass
filter is one whose passband doesn't undergo any change (i.e., no
gain nor attenuation), but completely rejects all frequencies
outside the passband. In an ideal band-pass filter, the
transition of the response from outside the passband to within the
passband and vice versa is instantaneous.
Of course, an ideal
band-pass filter doesn't exist in the real world, i.e., some
attenuation still occurs within the passband while complete
attenuation is not achieved outside the passband. The amount of
attenuation outside the passband may be described in terms of the
'roll-off'
of the filter, which is the attenuation in
dB per octave of frequency.
LC band-pass filters, or
filters containing resonant circuits composed of inductors and
capacitors, has a resonant frequency between the lower cut-off
frequency f1 and the upper cut-off frequency f2. At this
resonant frequency, the gain of the band pass filter is at its
maximum.
The
over-all impedance of a resonant series LC circuit consisting of an
inductor and a capacitor in series with each other will drop to zero
at the resonant frequency because the reactances of the inductor and
the capacitor cancel each other out under resonance. On the
other hand, the over-all impedance of a resonant parallel LC circuit
consisting of an inductor and a capacitor in parallel with each
other will increase to infinity at the resonant frequency, i.e., the
reactances of the inductor and the capacitor result in zero current
flow under resonance.
Resonant
series and parallel LC circuits may thus be combined to form a
band-pass filter as shown in Figure 1. In this circuit, the
resonant series LC circuits are used to allow only the desired
frequency range to pass while the resonant parallel LC circuit is
used to attenuate frequencies outside the passband by shunting them
towards the ground.
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Figure 1. A Band-Pass Filter |
The following equations apply to the band-pass
filter in Figure 1 above:
1) L = Zo / (π(f2-f1)); Lf = (f2-f1) /
(4π(f2f1))
2)
C = (f2-f1) / (4π(f2f1Zo)); Cf = 1 /
(π(f2-f1)Zo)
3) fo = sqrt(f1f2) = 1 / (2π(sqrt(LC))) = 1 /
(2π(sqrt(LfCf)))
4)
Zo = sqrt(L/Cf) = sqrt(Lf/C)
where Zo = line impedance; f1 = lower cut-off
frequency; f2 = upper cut-off frequency; fo =
resonant frequency.
See Also:
Band-Stop Filter;
Resonance;
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