- Building a passive first-order low-pass
or high-pass RC filter
is as simple as connecting a single resistor
in series with a capacitor.
When the input signal is applied to the resistor
and the output is measured across the capacitor
it creates a low-pass filter.
And when the location of the resistor and the capacitor
are smocked with each other
so that the output is measured
across the resistor,
it produces a high-pass filter.
The advantage of these passive circuits
is that they're easy to build,
it only takes two components,
and they're easy to design.
The equation for the cutoff frequency
is simply one over two Pi times the resistance
and the capacitance values.
Now, a major disadvantage of these passive RC circuits
is that since they only use passive components
the output signal can never be greater
than the input signal.
At the very best the input and output
can have the same power.
The attenuation can really start to add up
when we need to combine multiple filters together.
Additionally, the impedance of whatever load
is attached to that filter's output,
can affect its characteristics
and change the cutoff frequency.
To overcome those disadvantages
I'll convert the passive RC filter
into an active RC filter
by connecting an op-amp
to the output of the filter circuit.
The left side of this circuit
is just a basic low-pass RC filter
and I can use the same equation as before
to calculate its cutoff frequency.
This ten kiloohm resistor,
in series with a 100 nanofarad capacitor,
will have a cut-off frequency of around 159 hertz.
The output from the passive filter
is then fed into a unity gain buffer
on the right side of the circuit.
The op-amps exceptionally high input impedance
prevents it from overloading the filter's output.
So it doesn't really affect
the filter's frequency response.
And the op-amps low output impedance
enables it to be connected to a wide variety of loads.
Even though the output voltage from the unity gain buffer
will be the same as its input voltage
that buffer has the ability to increase and decrease
the power of the output signal
to respond to changes in the load impedance.
Since the op-amp is simply acting as a buffer
between the low-pass filter
and whatever load is attached to the output,
the frequency response for the circuit
should look similar to the response
for the passive RC filter.
The unity buffer configuration of the op-amp circuit
has a voltage gain of one,
which in terms of decibels is zero dB.
So down in the pass band region of this active filter
the output signal will have roughly the same
amplitude as the input signal.
If I want to make the output voltage larger than the input
then I can modify that circuit
to use an amplifying op-amp configuration
instead of a unity gain buffer.
In addition to serving as a buffer between the RC filter
and whatever load is attached to the output,
the non-inverting op-amp configuration shown here,
with two ten kiloohm resistors,
will amplify the filter's output by a factor of two.
In terms of decibels,
that voltage gain of two
corresponds to about a six dB increase.
So output signals with frequencies in the pass band
will have roughly twice the amplitude
of their corresponding input signals.
Keep in mind that since this is a low-pass filter
if the input signal has any zero-hertz DC offset,
that offset will also be amplified by the output.
If I swap around the location of the ten kiloohm resistor,
and the 100 nanofarad capacitor
in the RC circuit on the left,
then I've just converted my low-pass filter
into a high-pass filter
with the same cutoff frequency of around 159 hertz.
The frequency response for the active high-pass filter
should look like a flipped version
of the low-pass filter's frequency response.
And since it includes amplification,
frequencies within the filter's pass band
will experience a six dB voltage gain.
Now, in theory,
the pass band for a high-pass filter
should extend from the cut-off frequency up to infinity.
In reality, the op-amp will have its own frequency response.
That limits its open-loop voltage gain.
At some point
the op-amps frequency response will cross over
the high-pass RC filter's frequency response.
That's usually at a frequency up in the kilohertz range
or megahertz range depending on which op-amps you're using.
But above that cross-over point the op-amp will act
like a low-pass filter,
limiting the pass band.
Since there's no such thing as an op-amp
with infinite band width
it's not possible to create a true active high-pass filter
whose pass band goes up to infinity.
But as long as the signal frequencies
I'm interested in keeping
all exist within that pass band,
this filter should work just fine for me.
This is one reason to pay attention to the open-loop
frequency response when choosing which type of op-amp
to use for a certain application.
One of my favorite things about op-amps
is that their high input impedance
and low output impedance
make them great for impedance bridging.
So I can chain together several circuits,
passing a voltage signal from one to the next.
For example, I can use a unity gain buffer as a bridge
to connect the output from an RC low-pass filter
to the input of another RC high-pass filter.
In this circuit,
I've chosen to use a one kiloohm resistor
and a 100 nanofarad capacitor
for the low-pass filter on the left,
getting it a cut-off frequency of around 1592 hertz.
A unity gain buffer connects the output
from that low-pass filter
to another high-pass filter
whose cut-off frequency is only 159 hertz.
And I decided to add an amplifier stage
at the end of this chain
to boost the output voltage with the gain of 11.
Since that low-pass filter stage blocks signals
that are above 1592 hertz,
and the high-pass filter
blocks signals below 159 hertz,
together those two filters
connected together with op-amps
create a band pass filter
whose pass band spans from a range of 159 hertz
up to 1592 hertz.
And frequencies within that pass band
will be amplified with a gain of 11
which corresponds to about 20.8 decibels.
I've built that band pass filter's circuit
on my breadboard using a pair of op-amps
and I'll be using my function generator
to provide an input signal
to the beginning low-pass filter stage.
Right now the function generator is set
to produce a 100 millivolt peak-to-peak sine wave
with a frequency of one kilohertz,
which is shown in yellow on the oscilloscope.
Since one kilohertz is in the pass band
for both filter stages,
the input signal passes through the filter stages
more or less unaffected.
And the amplifier at the end
increases the amplitude of the output
shown in blue with a voltage gain of 11.
Now, if I lower the input frequency
down to around ten hertz
and adjust the horizontal scale on the scope
so I can view multiple periods of the signal at once,
I see that the output signal is much smaller
than it was a moment ago.
The ten hertz signal passes through
the first low-pass filter stage just fine,
but when it gets to the later high-pass filter stage,
it falls below the cutoff frequency.
So it gets attenuated.
I'll get a similar result
if I increase the input frequency
up to around ten kilohertz.
And adjust the horizontal scale on the oscilloscope
to get a better view.
Now the input signal is being attenuated
by that first low-pass stage in my band pass filter.
And any remaining energy left from that
passes on through the high-pass stage to the amplifier.