Filters
Electronic circuits can work over a wide range of frequencies, anywhere
from DC (zero hertz) to billions of cycles per second (gigahertz). It
is often
desirable to limit the range of frequencies that a circuit responds to.
Filters are used to limit the frequency response of a circuit.
A typical use of a filter circuit is a crossover network in a speaker
system. It is difficult to make a single loud speaker that responds to
the full range of audio frequencies. A woofer, a speaker that responds
to low
frequencies must be large and able to move a lot of air. This is just
the opposite of a tweeter, a speaker that responds to high frequencies.
It must be small and light so that it can vibrate at high frequencies.
The output of a high fidelity audio amplifier will respond to the full
range of audio frequencies. You could simply put a woofer and tweeter
in parallel. However, high frequencies reaching the woofer are wasted
energy. Low frequencies reaching the tweeter can cause it to overheat.
A crossover network is a filter that directs high frequencies to the
tweeter and low frequencies to the woofer. The circuit used in the
example below is not a practical crossover network because it has a
resistor as part of the filter. Power is produced in resistors which is
a waste energy. A practical crossover network is made of only
capacitors and inductors. Such LC filters will be seen later in this
section. Practical crossover networks will be discussed in analog
circuits.
RC Filters
In a circuit made of a single resistor
and a single capacitor there is
a frequency where the resistance and capacitive reactance will be
equal. This frequency is called the cutoff frequency
(ƒ_{CO}). Looking at the following
circuit, at frequencies below the cutoff frequency more voltage is
developed across the capacitor than the resistor. At frequencies above
the cutoff frequency more voltage is developed across the resistor
than the capacitor. If you were using this filter as a crossover
network you would want the tweeter across the resistor and the woofer
across the inductor.
An RC filter would not be used as a practical crossover network because
the resistor would waste power. RC filters are used in the low power
parts of a circuit, such as at microphone inputs and between
preamplifiers and power amplifiers.
Another definition of the cutoff frequency is the halfpower point. At
the cutoff frequency the power produced across the resistor and the
capacitor are equal. Therefore, 50% of the power is across each of the
two components. If the filter is seen as having an input and an output
(see below for further explanation), 50% of the power is reaching the
output.
Low pass RC Filter
In the above arrangement, the voltage across the resistor or the
capacitor can be isolated and used to connect to another circuit. For
example, the output of an audio preamplifier may be placed
across both components, the components seen as in series from the
preamplifier. The voltage developed across either the
resistor or the capacitor can be sent to the input of another amplifier
stage. The connection across both components is then considered the
input of the filter and the connection a single component is considered
the output of the filter.
If the connection across the capacitor is used as the output of the
filter a higher voltage will be developed at frequencies below the
cutoff frequency. Low frequencies will be passed through
the filter at a higher level than high frequencies. This arrangement
constitutes a low pass filter.
The following diagram shows a typical power output plot of a low pass
filter. The grid is laidout using a mathematical function called a
logarithm (discussed later in analog circuits). This distorts the
appearance of the plot in such a way that it appears as a flat line at
frequencies below the cutoff frequency and as a straight sloped line
at frequencies above the cutoff frequency. This makes a visual graph
that is easier to interpret than the continuouslycurved lines that
would appear in a linear function (with evenly spaced intervals) were
used. It also distorts the graph in such a way that the halfpower
point appears to be much higher than you might expect. The frequency
shown is relative; the 1000 doesn't necessarily represent 1,000 Hz. The
actual cutoff frequency depends on the components.
This plot illustrates the response of a low pass filter. The output is
unaffected by the filter at frequencies below the cutoff frequency but
drops off dramatically at frequencies above the cutoff frequency. This
particular plot drops of by a factor of 100 for every tenfold increase
in frequency. This would be referred to as 100 per decade.
Response of a low pass filter
The frequency is relative. The slope depends on the filter design. The
response of a typical filter is in red. The purple line is a bode plot,
which is a straightline approximation of a filter response. The grid
is plotted using a logarithmic function (decibels [dB]). This is why
the
halfpower point appears to be much greater than 50% of the maximum
level. Logarithms and decibels will be discussed in analog circuits.

Let's build a formula to find the cutoff frequency of an RC filter. At
the cutoff frequency the capacitive reactance equals the resistance.
X_{C} = R
Therefore, the resistance equals the formula for capacitive reactance:
_{R}_{
= }

1

——————— 
2πƒC

Using the rules for the basic algebraic formula we can pull the
frequency out and solve for it:
Since this only works at the cutoff frequency the frequency solved for
is the cutoff frequency. This gives
us the following formula to find the cutoff frequency:
ƒ_{CO}_{
= }

1

——————— 
2πRC 
Where:

ƒ_{CO}

=

cutoff
frequency 

2π 
=

Mathematical
constant of 6.28 

R

=

Resistance
in ohms 

C

=

Capacitance in
farads

For example, a 0.16 microfarad capacitor will have approximately 100
ohms of capacitive reactance at 10 kHz. If you make an RC filter using
this 0.16 microfarad capacitor and a 100 ohm resistor the cutoff
frequency will be 9,952Hz, which rounds to 10,000 Hz. χ
^{1}/χ
ƒ_{CO}_{
= }

1

≈ 10,000

————————————————————— 
6.28 x 100
x 0.00000016 
On the calculator that is:
6


.


2


8


X


1


0


0


X


.


0


0


0


0


0


0


1


6


=


^{1}_{/χ} 

9952...

An RC filter is typically illustrated with the following configuration.
Since the resistor and capacitor are drawn in the shape of the letter
“L” it is called an “L” filter.
An RC "L"
low pass filter

A low pass filter is used in audio circuits where more bass response is
desired. Radio circuits sometimes produce harmonic frequencies above
the desired radio frequency. A low pass filter may be used to pass the
desired frequency or range of frequencies while blocking the harmonics.
The ripple filter of a power supply (discussed in analog circuits) is a
low pass filter.
High pass RC Filter
The second circuit is the same as the low pass filter above except the
output is taken across the resistor.
This makes it a high pass filter. Everything else applies the same as
to the low pass filter. A high pass RC filter is typically drawn as
below. As above it is drawn in the shape of the letter "L" and is
called an "L" filter.
An RC high
pass "L" filter.
This drawing is "flipped" compared to the previous one, but the filter
works the same. The output is still across the resistor where the
higher voltage is developed at higher frequencies. 
Like the low pass filter, increasing the
value of the capacitor or the resistor will lower the cutoff
frequency. The frequency response plot for a high pass filter is a
"mirror image" of the plot for a low pass filter.
Response of a high pass filter
This is a mirror image of the
low pass filter graph.

High pass filters are used in audio circuits where more treble response
is desired or in radio circuits where unwanted frequencies are below
the desired frequencies.
LC Filters
LC filters have the advantage of not wasting energy as by dissipating
power in a resistor. Notice that in an LC filter you actually have a
resonant circuit.



LC
high pass "L" filter


LC
low pass "L" filter

Therefore, the cutoff frequency will be the resonant frequency. At
frequencies above the resonant frequency the inductive reactance is
greater than the capacitive reactance. This creates a voltage divider
where the voltage across the inductor is higher than the voltage across
the capacitor. At frequencies below the resonant frequency the
capacitive reactance is greater than the inductive reactance. The
voltage across the capacitor is higher than the voltage across the
inductor.
High pass filter
If the output is taken across the inductor of an LC filter you have a
high pass filter. Another way to look at it is that higher frequencies
pass through the capacitor to the output and lower frequencies go
through the inductor to ground, bypassing the output.
Low pass filter
If the output is taken across the capacitor of an LC filter you have a
low pass filter. Another way to look at it is that the lower
frequencies pass through the inductor to the output and the higher
frequencies go through the capacitor to ground, bypassing the output.
Band pass filter
A band pass filter is designed to pass a range of frequencies. A series
resonant circuit acts as a short circuit at the resonant
frequency so it passes the resonant frequency and blocks higher and
lower frequencies.
Input


Output

A series band pass filter
Frequencies above and below the resonant frequency are blocked.
Frequencies near the resonant frequency pass through. This filter is in
series with the signal.

A parallel resonant circuit has a very high impedance it the resonant
frequency. If you place a parallel resonant circuit across the signal
path frequencies above and below the resonant frequency will be shorted
out. Only frequencies near the resonant frequency will pass by the
circuit and reach the output.
Input


Output

A parallel band pass filter
Frequencies above and below the resonant frequency are shorted to
ground. Frequencies near the resonant frequency bypass the circuit.
This filter is parallel or "across" the signal. 
A band pass filter can be made with a series or parallel resonant
circuit. Which circuit is used depends on the orientation of the filter.
Bandreject filter
A bandreject filter (or notch filter) is designed to block a range of
frequencies. These are made the same way as a band pass filter except
for the orientation of the filter circuit. A parallel circuit in series
with the signal operates as a band reject filter.
Input


Output

A band reject filter using a parallel resonant circuit
Frequencies near the resonant frequency are blocked.
Frequencies above and below the resonant frequency pass through the
circuit. 
A series circuit across the signal path also operates as a band reject
filter.
Input


Output

A band reject filter using a series resonant circuit
Frequencies near the resonant frequency are shorted to ground.
Frequencies above and below the resonant frequency bypass the circuit. 
"Pi" Filters and "T" Filters
The previous filters are called “L” filters because the components are
usually illustrated with the components laid out in the shape of the
letter “L”. The “T” filter is usually illustrated with the components
in the shape of the letter “T”. “T” filters can have any combination of
components and be either low pass or high pass filters.
"T" filters



LC
high pass "T" filter


LC
low pass "T" filter

"Pi" Filters



LC
high pass "Pi" filter


LC
low pass "Pi" filter

Like the “T” filter, the Pi filter is usually illustrated with the
components laid out in the shape of the Greek letter Pi. Like the “T”
filter, the Pi can use any combination of components and be either
low pass or high pass. “T” and "Pi" filters are used where
impedance
matching between circuits is important. The filters can be designed to
couple circuits of different impedances.
"H" filters
"T" and "Pi" filters are unsymmetrical. One end is ties to ground and
the electrical signal they operate on is carried by a single wire (an
unbalanced signal). Often electrical signals are carried by pairs of
wires where the voltage in each wire is always opposite to the voltage
in the other. This is a balanced signal and requires symmetrical or
balanced filters. The most common balanced filter is the "H" filter.
This is a balanced version of the "T" filter and is drawn resembling
the letter "H".

LC
low pass "H" filter

Narrow band and wide band filters
In the section on resonant
circuits you can see that resonant circuits
can be designed to have a sharp peak (high Q) or a shallow peak (low Q
[review resonant circuits]). A high Q filter is a narrow band filter.
A high Q filter
The pass band of a high Q filter (the range of frequencies between the
cutoff frequencies) is narrow. The drop off above and below the pass
band is steep; the response of the filter drops off rapidly as you go
above and below the pass band.

The sharp drop off (steep skirts) of a high Q filter is often
desirable, but it only passes a narrow range of frequencies. To pass a
wide range of frequencies you need a low Q filter.
A low Q filter
The pass band of a low Q filter is wide. However, the drop off above
and below the pass
band is shallow; the response of the filter does not drop off as much
as you go
above and below the pass band.

Low Q filters have an inherent problem. Without the seep skirts they
don't reject unwanted frequencies very well.
What do you do if you want the steep drop off of a high Q filter and
the wide band of a low Q filter?
Filters can be designed that are essentially multiple filters in one
unit as follows:
A Compound Filter
This is three bandpass filters in one. It is essentially two parallel
bandpass filters with a series bandpass filter in between.

The three stages in the above filter are tuned to
slightly different frequencies. The first stage passes one range of
frequencies while the middle stage passes another range that overlaps
the first. The third state passes yet another range of frequencies.
Overlapping
Stage Responses

The overall response of a compound filter is something like
this.
A Compound Filter's Overall
Response
The stages of a compound filter overlap to give an overall response
something like this diagram

There are several design types for compound filters. Some of these are
the Chebychev filter, the Butterworth filter and the Constant K filter.
Filters may also be made as part of the feedback loops of amplifiers.
These are called active filters and have improved characteristics.
Compound filter design is an engineering field and is beyond the scope
of this course.