When both types of filters are combined into one, that is, a capacitor & resistor in series is used as input & a capacitor & resistor are
used in parallel for the feedback, a new type of filter emerges: the
bandpass filter.
To see how this works, we need to simplify the circuit to use only one
element instead of two, in order to make analysis easier. Since when AC
is applied to a capacitor it can be replaced with its capacitive
reactance in ohms, we can use that to combine it with the series
resistor at the input, & with the parallel resistor for feedback.
This gives us an input impedance (Impedance is a generalization of
resistance that also includes reactances, & is also measured in ohms) & a feedback impedance, in a configuration similar to the simple
inverting amplifier.
Since both impedances are frequency dependent, the gain will be
frequency dependent as well. At low frequencies, the input capacitor's
reactance is very high and dominates the series combination with the
resistor, so the input impedance becomes very large. At the same time,
the feedback capacitor will also have a very high reactance, but this
time the resistor dominates because the connection is made in parallel.
Since the gain is defined by the ratio -Rf/Rin, generalized to
impedances as -Zf/Zin, where Z denominates impedances in most
electronics literature. Since the feedback impedance is small, limited
by the resistor, compared the input impedance which tends to infinity,
the ratio will be very small & will attenuate the signal (Zf <<
Zin, so the ratio is less than 1). In this case, the extremely high
input impedance drives the ratio towards zero.
At very high frequencies, the input impedance is dominated by the
resistance, since the capacitor's reactance is very small. The opposite
effect happens at the feedback, since now the capacitor dominates with
its very low reactance, which makes the impedance very low.
Checking the gain ratio -Zf/Zin, we can see that now the input impedance
is very low, limited by the input resistor, but the feedback impedance
will be lower still, going towards zero, not being limited by anything
since the capacitor is dominating the connection, so the ratio will
again be very small, attenuating the signal. This time, the very small
feedback impedance drives the ratio to zero.
At medium frequencies, where no single component dominates each
connection, both input & feedback impedance will be very close to each
other, since they will be a very similar value, assuming equal
components. At the frequency where the series combination & the
parallel combination have the same value, the gain will be 1, given by
the ratio -Zf/Zin, where Zf = Zin; This is called the center frequency, & it is the only signal that will not be attenuated.
The overall effect is that this circuit will attenuate both high & low
frequency signals applied to it, & only pass a small range (also
called band) of frequencies where both input & feedback impedances
have a very similar value, hence the name bandpass filter. This is
useful when you need to block noise or extra signals created within a
circuit.
