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We've looked at low-pass filters and high-pass 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 high-pass filter, while R2 and C2 form a low-pass filter. For the purpose of discussion, we arbitrarily assign a cutoff frequency ωCO1 = 10 radians/sec for R1 and C1, and a higher cutoff frequency ωCO2 = 10,000 radians/sec for R2 and C2. The actual frequencies don't matter, so long as ωCO1 is less than ωCO2. 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 low-pass filter first. However, if we use the circuit shown here, the dc resistance between vOUT and ground is R1 + R2. If we swap the two filters, R1 will be the only resistance from vOUT to ground. In addition, the second filter section will present a load to the first section. Since the low-pass section has a higher cutoff frequency (ωCO2), 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 low-frequency cutoff, and will block signals at lower frequencies while passing higher-frequency signals.
These signals will also be passed by R2 and C2, so long as their frequency doesn't get too high. Frequencies above ωCO2 pass through C2 to ground, and therefore are kept away from vOUT.
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 band-pass 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 ωCO1 is set equal to ωCO2. Then the response curve peaks at the mutual cutoff frequency and rolls off immediately on either side.
If you attempt to set ωCO1 to a higher frequency than ωCO2, the band-pass filter will block all frequencies, and no signal will get through.
Because the band-pass filter is actually two independent first-order 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 high-pass section cancels the phase lag from the low-pass 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 band-pass filter. In our example circuit, the high-pass filter comes first, and has its effect on the signal. vOUT from the high-pass filter becomes vIN for the low-pass 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 high-pass filter first, since the low-pass filter, with a higher cutoff frequency, will have higher relative impedance values for its components.
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