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BandPass Filters 

We've looked at lowpass filters and highpass filters individually. But what happens if we combine them into a single circuit, as shown to the right?
In this case, R1 and C1 form a highpass filter, while R2 and C2 form a lowpass filter. For the purpose of discussion, we arbitrarily assign a cutoff frequency ω_{CO1} = 10 radians/sec for R1 and C1, and a higher cutoff frequency ω_{CO2} = 10,000 radians/sec for R2 and C2. The actual frequencies don't matter, so long as ω_{CO1} is less than ω_{CO2}. That way, R1 and C1 pass signals that will also be passed by R2 and C2.
It is equally possible to swap the two filter sections, putting the lowpass filter first. However, if we use the circuit shown here, the dc resistance between v_{OUT} and ground is R1 + R2. If we swap the two filters, R1 will be the only resistance from v_{OUT} to ground. In addition, the second filter section will present a load to the first section. Since the lowpass section has a higher cutoff frequency (ω_{CO2}), R2 and C2 have higher impedances and constitute less of a load on R1 and C1 than would be true if the sections were swapped. Therefore the two filters operate pretty much independently, even though they are electrically connected.
If we apply the cutoff frequencies assumed above, the frequency response curve for our filter will appear as shown to the right. R1 and C1 govern the lowfrequency cutoff, and will block signals at lower frequencies while passing higherfrequency signals.
These signals will also be passed by R2 and C2, so long as their frequency doesn't get too high. Frequencies above ω_{CO2} pass through C2 to ground, and therefore are kept away from v_{OUT}.
The actual band or range of frequencies passed by this type of filter does not have to cover three decades as shown here. The two parts of the bandpass filter can be adjusted independently of each other to widen or narrow the pass band as much as you like. The minimum effective pass band occurs when ω_{CO1} is set equal to ω_{CO2}. Then the response curve peaks at the mutual cutoff frequency and rolls off immediately on either side.
If you attempt to set ω_{CO1} to a higher frequency than ω_{CO2}, the bandpass filter will block all frequencies, and no signal will get through.
Because the bandpass filter is actually two independent firstorder filters, the phase response of the entire circuit is simply the combination of the phase responses of the two separate sections. This combined phase response is shown in the graph to the right.
In this case, the pass band is only three decades wide, so the output phase shift is zero only for a very narrow range of frequencies. A wider pass band would mean a correspondingly wider frequency range with no phase shift. A narrower pass band results in a narrower frequency range with no phase shift.
In the case where the two filter sections have the same cutoff frequency, the phase lead from the highpass section cancels the phase lag from the lowpass section at the cutoff frequency only, so that is the only frequency with no phase shift.
There are no special extra calculations required for the bandpass filter. In our example circuit, the highpass filter comes first, and has its effect on the signal. v_{OUT} from the highpass filter becomes v_{IN} for the lowpass filter, which then has its effect on the signal. The two filters don't really interact with each other, beyond the fact that the second filter, depending on component values, may act as a load on the first one. To minimize this effect, we generally put the highpass filter first, since the lowpass filter, with a higher cutoff frequency, will have higher relative impedance values for its components.


 
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