What is a filter?
It is sometimes desirable to have
circuits capable of selectively filtering one frequency or range of
frequencies out of a mix of different frequencies in a circuit. A
circuit designed to perform this frequency selection is called a
filter circuit, or simply a filter. A common need for filter
circuits is in high-performance stereo systems, where certain ranges of
audio frequencies need to be amplified or suppressed for best sound
quality and power efficiency. You may be familiar with equalizers,
which allow the amplitudes of several frequency ranges to be adjusted to
suit the listener's taste and acoustic properties of the listening area.
You may also be familiar with crossover networks, which block
certain ranges of frequencies from reaching speakers. A tweeter
(high-frequency speaker) is inefficient at reproducing low-frequency
signals such as drum beats, so a crossover circuit is connected between
the tweeter and the stereo's output terminals to block low-frequency
signals, only passing high-frequency signals to the speaker's connection
terminals. This gives better audio system efficiency and thus better
performance. Both equalizers and crossover networks are examples of
filters, designed to accomplish filtering of certain frequencies.
Another practical application of filter
circuits is in the "conditioning" of non-sinusoidal voltage waveforms in
power circuits. Some electronic devices are sensitive to the presence of
harmonics in the power supply voltage, and so require power conditioning
for proper operation. If a distorted sine-wave voltage behaves like a
series of harmonic waveforms added to the fundamental frequency, then it
should be possible to construct a filter circuit that only allows the
fundamental waveform frequency to pass through, blocking all
(higher-frequency) harmonics.
We will be studying the design of several
elementary filter circuits in this lesson. To reduce the load of math on
the reader, I will make extensive use of SPICE as an analysis tool,
displaying Bode plots (amplitude versus frequency) for the various kinds
of filters. Bear in mind, though, that these circuits can be analyzed
over several points of frequency by repeated series-parallel analysis,
much like the previous example with two sources (60 and 90 Hz), if the
student is willing to invest a lot of time working and re-working
circuit calculations for each frequency.
- REVIEW:
- A filter is an AC circuit that
separates some frequencies from others in within mixed-frequency
signals.
- Audio equalizers and
crossover networks are two well-known applications of filter
circuits.
- A Bode plot is a graph plotting
waveform amplitude or phase on one axis and frequency on the other.
Low-pass filters
By definition, a low-pass filter is a
circuit offering easy passage to low-frequency signals and difficult
passage to high-frequency signals. There are two basic kinds of circuits
capable of accomplishing this objective, and many variations of each
one:
The inductor's impedance increases with
increasing frequency. This high impedance in series tends to block
high-frequency signals from getting to the load. This can be
demonstrated with a SPICE analysis:
inductive lowpass filter
v1 1 0 ac 1 sin
l1 1 2 3
rload 2 0 1k
.ac lin 20 1 200
.plot ac v(2)
.end
freq v(2) 0.2512 0.3981 0.631 1
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+00 9.998E-01 . . . *
1.147E+01 9.774E-01 . . . *.
2.195E+01 9.240E-01 . . . * .
3.242E+01 8.533E-01 . . . * .
4.289E+01 7.776E-01 . . . * .
5.337E+01 7.050E-01 . . . * .
6.384E+01 6.391E-01 . . * .
7.432E+01 5.810E-01 . . * . .
8.479E+01 5.304E-01 . . * . .
9.526E+01 4.865E-01 . . * . .
1.057E+02 4.485E-01 . . * . .
1.162E+02 4.153E-01 . .* . .
1.267E+02 3.863E-01 . *. . .
1.372E+02 3.607E-01 . * . . .
1.476E+02 3.382E-01 . * . . .
1.581E+02 3.181E-01 . * . . .
1.686E+02 3.002E-01 . * . . .
1.791E+02 2.841E-01 . * . . .
1.895E+02 2.696E-01 . * . . .
2.000E+02 2.564E-01 .* . . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Load voltage decreases with increasing frequency
The capacitor's impedance decreases with
increasing frequency. This low impedance in parallel with the load
resistance tends to short out high-frequency signals, dropping most of
the voltage gets across series resistor R1.
capacitive lowpass filter
v1 1 0 ac 1 sin
r1 1 2 500
c1 2 0 7u
rload 2 0 1k
.ac lin 20 30 150
.plot ac v(2)
.end
freq v(2) 0.3162 0.3981 0.5012 0.631
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3.000E+01 6.102E-01 . . . . *.
3.632E+01 5.885E-01 . . . . * .
4.263E+01 5.653E-01 . . . . * .
4.895E+01 5.416E-01 . . . . * .
5.526E+01 5.180E-01 . . . .* .
6.158E+01 4.948E-01 . . . *. .
6.789E+01 4.725E-01 . . . * . .
7.421E+01 4.511E-01 . . . * . .
8.053E+01 4.309E-01 . . . * . .
8.684E+01 4.118E-01 . . .* . .
9.316E+01 3.938E-01 . . *. . .
9.947E+01 3.770E-01 . . * . . .
1.058E+02 3.613E-01 . . * . . .
1.121E+02 3.465E-01 . . * . . .
1.184E+02 3.327E-01 . .* . . .
1.247E+02 3.199E-01 . * . . .
1.311E+02 3.078E-01 . * . . . .
1.374E+02 2.965E-01 . * . . . .
1.437E+02 2.859E-01 . * . . . .
1.500E+02 2.760E-01 .* . . . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Load voltage decreases with increasing frequency
The inductive low-pass filter is the
pinnacle of simplicity, with only one component comprising the filter.
The capacitive version of this filter is not that much more complex,
with only a resistor and capacitor needed for operation. However,
despite their increased complexity, capacitive filter designs are
generally preferred over inductive because capacitors tend to be "purer"
reactive components than inductors and therefore are more predictable in
their behavior. By "pure" I mean that capacitors exhibit little
resistive effects than inductors, making them almost 100% reactive.
Inductors, on the other hand, typically exhibit significant dissipative
(resistor-like) effects, both in the long lengths of wire used to make
them, and in the magnetic losses of the core material. Capacitors also
tend to participate less in "coupling" effects with other components
(generate and/or receive interference from other components via mutual
electric or magnetic fields) than inductors, and are less expensive.
However, the inductive low-pass filter is
often preferred in AC-DC power supplies to filter out the AC "ripple"
waveform created when AC is converted (rectified) into DC, passing only
the pure DC component. The primary reason for this is the requirement of
low filter resistance for the output of such a power supply. A
capacitive low-pass filter requires an extra resistance in series with
the source, whereas the inductive low-pass filter does not. In the
design of a high-current circuit like a DC power supply where additional
series resistance is undesirable, the inductive low-pass filter is the
better design choice. On the other hand, if low weight and compact size
are higher priorities than low internal supply resistance in a power
supply design, the capacitive low-pass filter might make more sense.
All low-pass filters are rated at a
certain cutoff frequency. That is, the frequency above which the
output voltage falls below 70.7% of the input voltage. This cutoff
percentage of 70.7 is not really arbitrary, all though it may seem so at
first glance. In a simple capacitive/resistive low-pass filter, it is
the frequency at which capacitive reactance in ohms equals resistance in
ohms. In a simple capacitive low-pass filter (one resistor, one
capacitor), the cutoff frequency is given as:
Inserting the values of R and C from the
last SPICE simulation into this formula, we arrive at a cutoff frequency
of 45.473 Hz. However, when we look at the plot generated by the SPICE
simulation, we see the load voltage well below 70.7% of the source
voltage (1 volt) even at a frequency as low as 30 Hz, below the
calculated cutoff point. What's wrong? The problem here is that the load
resistance of 1 kΩ affects the frequency response of the filter, skewing
it down from what the formula told us it would be. Without that load
resistance in place, SPICE produces a Bode plot whose numbers make more
sense:
capacitive lowpass filter
v1 1 0 ac 1 sin
r1 1 2 500
c1 2 0 7u
* note: no load resistor!
.ac lin 20 40 50
.plot ac v(2)
.end
freq v(2) 0.6607 0.6918 0.7244 0.7586
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
4.000E+01 7.508E-01 . . . * .
4.053E+01 7.465E-01 . . . * .
4.105E+01 7.423E-01 . . . * .
4.158E+01 7.380E-01 . . . * .
4.211E+01 7.338E-01 . . . * .
4.263E+01 7.295E-01 . . . * .
4.316E+01 7.253E-01 . . * .
4.368E+01 7.211E-01 . . *. .
4.421E+01 7.170E-01 . . * . .
4.474E+01 7.129E-01 . . * . .
4.526E+01 7.087E-01 . . * . .
4.579E+01 7.046E-01 . . * . .
4.632E+01 7.006E-01 . . * . .
4.684E+01 6.965E-01 . . * . .
4.737E+01 6.925E-01 . * . .
4.789E+01 6.885E-01 . *. . .
4.842E+01 6.846E-01 . * . . .
4.895E+01 6.806E-01 . * . . .
4.947E+01 6.767E-01 . * . . .
5.000E+01 6.728E-01 . * . . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
At 45.26 Hz, the output voltage is above 70.7 percent;
At 45.79 Hz, the output voltage is below 70.7 percent;
It should be exactly 70.7% at 45.473 Hz!
When dealing with filter circuits, it is
always important to note that the response of the filter depends on the
filter's component values and the impedance of the load. If a
cutoff frequency equation fails to give consideration to load impedance,
it assumes no load and will fail to give accurate results for a
real-life filter conducting power to a load.
One frequent application of the
capacitive low-pass filter principle is in the design of circuits having
components or sections sensitive to electrical "noise." As mentioned at
the beginning of the last chapter, sometimes AC signals can "couple"
from one circuit to another via capacitance (Cstray) and/or
mutual inductance (Mstray) between the two sets of
conductors. A prime example of this is unwanted AC signals ("noise")
becoming impressed on DC power lines supplying sensitive circuits:
The oscilloscope-meter on the left shows
the "clean" power from the DC voltage source. After coupling with the AC
noise source via stray mutual inductance and stray capacitance, though,
the voltage as measured at the load terminals is now a mix of AC and DC,
the AC being unwanted. Normally, one would expect Eload to be
precisely identical to Esource, because the uninterrupted
conductors connecting them should make the two sets of points
electrically common. However, power conductor impedance allows the two
voltages to differ, which means the noise magnitude can vary at
different points in the DC system.
If we wish to prevent such "noise" from
reaching the DC load, all we need to do is connect a low-pass filter
near the load to block any coupled signals. In its simplest form, this
is nothing more than a capacitor connected directly across the power
terminals of the load, the capacitor behaving as a very low impedance to
any AC noise, and shorting it out. Such a capacitor is called a
decoupling capacitor:
A cursory glance at a crowded
printed-circuit board (PCB) will typically reveal decoupling capacitors
scattered throughout, usually located as close as possible to the
sensitive DC loads. Capacitor size is usually 0.1 µF or more, a minimum
amount of capacitance needed to produce a low enough impedance to short
out any noise. Greater capacitance will do a better job at filtering
noise, but size and economics limit decoupling capacitors to meager
values.
- REVIEW:
- A low-pass filter allows for easy
passage of low-frequency signals from source to load, and difficult
passage of high-frequency signals.
- Inductive low-pass filters insert an
inductor in series with the load; capacitive low-pass filters insert a
resistor in series and a capacitor in parallel with the load. The
former filter design tries to "block" the unwanted frequency signal
while the latter tries to short it out.
- The cutoff frequency for a
low-pass filter is that frequency at which the output (load) voltage
equals 70.7% of the input (source) voltage. Above the cutoff
frequency, the output voltage is lower than 70.7% of the input, and
visa-versa.
High-pass filters
A high-pass filter's task is just the
opposite of a low-pass filter: to offer easy passage of a high-frequency
signal and difficult passage to a low-frequency signal. As one might
expect, the inductive and capacitive versions of the high-pass filter
are just the opposite of their respective low-pass filter designs:
The capacitor's impedance increases with
decreasing frequency. This high impedance in series tends to block
low-frequency signals from getting to load.
capacitive highpass filter
v1 1 0 ac 1 sin
c1 1 2 0.5u
rload 2 0 1k
.ac lin 20 1 200
.plot ac v(2)
.end
freq v(2) 1.000E-03 1.000E-02 1.000E-01 1.000E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+00 3.142E-03 . * . . .
1.147E+01 3.602E-02 . . * . .
2.195E+01 6.879E-02 . . * . .
3.242E+01 1.013E-01 . . * .
4.289E+01 1.336E-01 . . . * .
5.337E+01 1.654E-01 . . . * .
6.384E+01 1.966E-01 . . . * .
7.432E+01 2.274E-01 . . . * .
8.479E+01 2.574E-01 . . . * .
9.526E+01 2.867E-01 . . . * .
1.057E+02 3.152E-01 . . . * .
1.162E+02 3.429E-01 . . . * .
1.267E+02 3.698E-01 . . . * .
1.372E+02 3.957E-01 . . . * .
1.476E+02 4.207E-01 . . . * .
1.581E+02 4.448E-01 . . . * .
1.686E+02 4.680E-01 . . . * .
1.791E+02 4.903E-01 . . . * .
1.895E+02 5.116E-01 . . . * .
2.000E+02 5.320E-01 . . . * .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Load voltage increases with increasing frequency
The inductor's impedance decreases with
decreasing frequency. This low impedance in parallel tends to short out
low-frequency signals from getting to the load resistor. As a
consequence, most of the voltage gets dropped across series resistor R1.
inductive highpass filter
v1 1 0 ac 1 sin
r1 1 2 200
l1 2 0 100m
rload 2 0 1k
.ac lin 20 1 200
.plot ac v(2)
.end
freq v(2) 1.000E-03 1.000E-02 1.000E-01 1.000E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+00 3.142E-03 . * . . .
1.147E+01 3.601E-02 . . * . .
2.195E+01 6.871E-02 . . * . .
3.242E+01 1.011E-01 . . * .
4.289E+01 1.330E-01 . . . * .
5.337E+01 1.644E-01 . . . * .
6.384E+01 1.950E-01 . . . * .
7.432E+01 2.248E-01 . . . * .
8.479E+01 2.537E-01 . . . * .
9.526E+01 2.817E-01 . . . * .
1.057E+02 3.086E-01 . . . * .
1.162E+02 3.344E-01 . . . * .
1.267E+02 3.591E-01 . . . * .
1.372E+02 3.828E-01 . . . * .
1.476E+02 4.053E-01 . . . * .
1.581E+02 4.267E-01 . . . * .
1.686E+02 4.470E-01 . . . * .
1.791E+02 4.662E-01 . . . * .
1.895E+02 4.845E-01 . . . * .
2.000E+02 5.017E-01 . . . * .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Load voltage increases with increasing frequency
This time, the capacitive design is the
simplest, requiring only one component above and beyond the load. And,
again, the reactive purity of capacitors over inductors tends to favor
their use in filter design, especially with high-pass filters where high
frequencies commonly cause inductors to behave strangely due to the skin
effect and electromagnetic core losses.
As with low-pass filters, high-pass
filters have a rated cutoff frequency, above which the output
voltage increases above 70.7% of the input voltage. Just as in the case
of the capacitive low-pass filter circuit, the capacitive high-pass
filter's cutoff frequency can be found with the same formula:
In the example circuit, there is no
resistance other than the load resistor, so that is the value for R in
the formula.
Using a stereo system as a practical
example, a capacitor connected in series with the tweeter (treble)
speaker will serve as a high-pass filter, imposing a high impedance to
low-frequency bass signals, thereby preventing that power from being
wasted on a speaker inefficient for reproducing such sounds. In like
fashion, an inductor connected in series with the woofer (bass) speaker
will serve as a low-pass filter for the low frequencies that particular
speaker is designed to reproduce. In this simple example circuit, the
midrange speaker is subjected to the full spectrum of frequencies from
the stereo's output. More elaborate filter networks are sometimes used,
but this should give you the general idea. Also bear in mind that I'm
only showing you one channel (either left or right) on this stereo
system. A real stereo would have six speakers: 2 woofers, 2 midranges,
and 2 tweeters.
For better performance yet, we might like
to have some kind of filter circuit capable of passing frequencies that
are between low (bass) and high (treble) to the midrange speaker so that
none of the low- or high-frequency signal power is wasted on a speaker
incapable of efficiently reproducing those sounds. What we would be
looking for is called a band-pass filter, which is the topic of
the next section.
- REVIEW:
- A high-pass filter allows for easy
passage of high-frequency signals from source to load, and difficult
passage of low-frequency signals.
- Capacitive high-pass filters insert a
capacitor in series with the load; inductive high-pass filters insert
a resistor in series and an inductor in parallel with the load. The
former filter design tries to "block" the unwanted frequency signal
while the latter tries to short it out.
- The cutoff frequency for a
high-pass filter is that frequency at which the output (load) voltage
equals 70.7% of the input (source) voltage. Above the cutoff
frequency, the output voltage is greater than 70.7% of the input, and
visa-versa.
Band-pass filters
There are applications where a particular
band, or spread, or frequencies need to be filtered from a wider range
of mixed signals. Filter circuits can be designed to accomplish this
task by combining the properties of low-pass and high-pass into a single
filter. The result is called a band-pass filter. Creating a
bandpass filter from a low-pass and high-pass filter can be illustrated
using block diagrams:
What emerges from the series combination
of these two filter circuits is a circuit that will only allow passage
of those frequencies that are neither too high nor too low. Using real
components, here is what a typical schematic might look like:
capacitive bandpass filter
v1 1 0 ac 1 sin
r1 1 2 200
c1 2 0 2.5u
c2 2 3 1u
rload 3 0 1k
.ac lin 20 100 500
.plot ac v(3)
.end
freq v(3) 4.467E-01 5.012E-01 5.623E-01 6.310E-01
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 4.703E-01 . * . . .
1.211E+02 5.155E-01 . . * . .
1.421E+02 5.469E-01 . . * . .
1.632E+02 5.676E-01 . . .* .
1.842E+02 5.801E-01 . . . * .
2.053E+02 5.865E-01 . . . * .
2.263E+02 5.882E-01 . . . * .
2.474E+02 5.864E-01 . . . * .
2.684E+02 5.820E-01 . . . * .
2.895E+02 5.755E-01 . . . * .
3.105E+02 5.676E-01 . . .* .
3.316E+02 5.585E-01 . . *. .
3.526E+02 5.487E-01 . . * . .
3.737E+02 5.384E-01 . . * . .
3.947E+02 5.277E-01 . . * . .
4.158E+02 5.169E-01 . . * . .
4.368E+02 5.060E-01 . .* . .
4.579E+02 4.951E-01 . *. . .
4.789E+02 4.843E-01 . * . . .
5.000E+02 4.736E-01 . * . . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Load voltage peaks within narrow frequency range
Band-pass filters can also be constructed
using inductors, but as mentioned before, the reactive "purity" of
capacitors gives them a design advantage. If we were to design a
bandpass filter using inductors, it might look something like this:
The fact that the high-pass section comes
"first" in this design instead of the low-pass section makes no
difference in its overall operation. It will still filter out all
frequencies too high or too low.
While the general idea of combining
low-pass and high-pass filters together to make a bandpass filter is
sound, it is not without certain limitations. Because this type of
band-pass filter works by relying on either section to block
unwanted frequencies, it can be difficult to design such a filter to
allow unhindered passage within the desired frequency range. Both the
low-pass and high-pass sections will always be blocking signals to some
extent, and their combined effort makes for an attenuated (reduced
amplitude) signal at best, even at the peak of the "pass-band" frequency
range. Notice the curve peak on the previous SPICE analysis: the load
voltage of this filter never rises above 0.59 volts, although the source
voltage is a full volt. This signal attenuation becomes more pronounced
if the filter is designed to be more selective (steeper curve, narrower
band of passable frequencies).
There are other methods to achieve
band-pass operation without sacrificing signal strength within the
pass-band. We will discuss those methods a little later in this chapter.
- REVIEW:
- A band-pass filter works to
screen out frequencies that are too low or too high, giving easy
passage only to frequencies within a certain range.
- Band-pass filters can be made by
stacking a low-pass filter on the end of a high-pass filter, or
visa-versa.
- "Attenuate" means to reduce or
diminish in amplitude. When you turn down the volume control on your
stereo, you are "attenuating" the signal being sent to the speakers.
Band-stop filters
Also called band-elimination,
band-reject, or notch filters, this kind of filter passes all
frequencies above and below a particular range set by the component
values. Not surprisingly, it can be made out of a low-pass and a
high-pass filter, just like the band-pass design, except that this time
we connect the two filter sections in parallel with each other instead
of in series.
Constructed using two capacitive filter
sections, it looks something like this:
The low-pass filter section is comprised
of R1, R2, and C1 in a "T"
configuration. The high-pass filter section is comprised of C2,
C3, and R3 in a "T' configuration as well.
Together, this arrangement is commonly known as a "Twin-T" filter,
giving sharp response when the component values are chosen in the
following ratios:
Given these component ratios, the
frequency of maximum rejection (the "notch frequency") can be calculated
as follows:
The impressive band-stopping ability of
this filter is illustrated by the following SPICE analysis:
twin-t bandstop filter
v1 1 0 ac 1 sin
r1 1 2 200
c1 2 0 2u
r2 2 3 200
c2 1 4 1u
r3 4 0 100
c3 4 3 1u
rload 3 0 1k
.ac lin 20 200 1.5k
.plot ac v(3)
.end
freq v(3) 1.000E-02 3.162E-02 1.000E-01 3.162E-01
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
2.000E+02 5.400E-01 . . . . *.
2.684E+02 4.512E-01 . . . . * .
3.368E+02 3.686E-01 . . . . * .
4.053E+02 2.946E-01 . . . *. .
4.737E+02 2.290E-01 . . . * . .
5.421E+02 1.707E-01 . . . * . .
6.105E+02 1.185E-01 . . . * . .
6.789E+02 7.134E-02 . . * . . .
7.474E+02 2.832E-02 . *. . . .
8.158E+02 1.126E-02 .* . . . .
8.842E+02 4.796E-02 . . * . . .
9.526E+02 8.222E-02 . . * . . .
1.021E+03 1.144E-01 . . . * . .
1.089E+03 1.447E-01 . . . * . .
1.158E+03 1.734E-01 . . . * . .
1.226E+03 2.007E-01 . . . * . .
1.295E+03 2.267E-01 . . . * . .
1.363E+03 2.515E-01 . . . * . .
1.432E+03 2.752E-01 . . . * . .
1.500E+03 2.980E-01 . . . *. .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- REVIEW:
- A band-stop filter works to
screen out frequencies that are within a certain range, giving easy
passage only to frequencies outside of that range. Also known as
band-elimination, band-reject, or notch filters.
- Band-stop filters can be made by
placing a low-pass filter in parallel with a high-pass filter.
Commonly, both the low-pass and high-pass filter sections are of the
"T" configuration, giving the name "Twin-T" to the band-stop
combination.
- The frequency of maximum attenuation
is called the notch frequency.
Resonant filters
So far, the filter designs we've
concentrated on have employed either capacitors or
inductors, but never both at the same time. We should know by now that
combinations of L and C will tend to resonate, and this property can be
exploited in designing band-pass and band-stop filter circuits.
Series LC circuits give minimum impedance
at resonance, while parallel LC ("tank") circuits give maximum impedance
at their resonant frequency. Knowing this, we have two basic strategies
for designing either band-pass or band-stop filters.
For band-pass filters, the two basic
resonant strategies are this: series LC to pass a signal, or parallel LC
to short a signal. The two schemes will be contrasted and simulated
here:
Series LC components pass signal at
resonance, and block signals of any other frequencies from getting to
the load.
series resonant bandpass filter
v1 1 0 ac 1 sin
l1 1 2 1
c1 2 3 1u
rload 3 0 1k
.ac lin 20 50 250
.plot ac v(3)
.end
freq v(3) 2.512E-01 3.981E-01 6.310E-01 1.000E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
5.000E+01 3.291E-01 . * . . .
6.053E+01 4.063E-01 . .* . .
7.105E+01 4.870E-01 . . * . .
8.158E+01 5.708E-01 . . * . .
9.211E+01 6.564E-01 . . .* .
1.026E+02 7.411E-01 . . . * .
1.132E+02 8.210E-01 . . . * .
1.237E+02 8.910E-01 . . . * .
1.342E+02 9.460E-01 . . . * .
1.447E+02 9.824E-01 . . . *.
1.553E+02 9.988E-01 . . . *
1.658E+02 9.967E-01 . . . *
1.763E+02 9.796E-01 . . . *.
1.868E+02 9.518E-01 . . . * .
1.974E+02 9.174E-01 . . . * .
2.079E+02 8.797E-01 . . . * .
2.184E+02 8.408E-01 . . . * .
2.289E+02 8.026E-01 . . . * .
2.395E+02 7.657E-01 . . . * .
2.500E+02 7.307E-01 . . . * .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Load voltage peaks at resonant frequency (159.15 Hz)
A couple of points to note: see how there
is virtually no signal attenuation within the "pass band" (the range of
frequencies near the load voltage peak), unlike the band-pass filters
made from capacitors or inductors alone. Also, since this filter works
on the principle of series LC resonance, the resonant frequency of which
is unaffected by circuit resistance, the value of the load resistor will
not skew the peak frequency. However, different values for the load
resistor will change the "steepness" of the Bode plot (the
"selectivity" of the filter).
The other basic style of resonant
band-pass filters employs a tank circuit (parallel LC combination) to
short out signals too high or too low in frequency from getting to the
load:
The tank circuit will have a lot of
impedance at resonance, allowing the signal to get to the load with
minimal attenuation. Under or over resonant frequency, however, the tank
circuit will have a low impedance, shorting out the signal and dropping
most of it across series resistor R1.
parallel resonant bandpass filter
v1 1 0 ac 1 sin
r1 1 2 500
l1 2 0 100m
c1 2 0 10u
rload 2 0 1k
.ac lin 20 50 250
.plot ac v(2)
.end
freq v(2) 3.162E-02 1.000E-01 3.162E-01 1.000E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
5.000E+01 6.933E-02 . * . . .
6.053E+01 8.814E-02 . * . . .
7.105E+01 1.100E-01 . .* . .
8.158E+01 1.361E-01 . . * . .
9.211E+01 1.684E-01 . . * . .
1.026E+02 2.096E-01 . . * . .
1.132E+02 2.640E-01 . . * . .
1.237E+02 3.382E-01 . . .* .
1.342E+02 4.392E-01 . . . * .
1.447E+02 5.630E-01 . . . * .
1.553E+02 6.578E-01 . . . * .
1.658E+02 6.432E-01 . . . * .
1.763E+02 5.503E-01 . . . * .
1.868E+02 4.543E-01 . . . * .
1.974E+02 3.792E-01 . . . * .
2.079E+02 3.234E-01 . . * .
2.184E+02 2.816E-01 . . *. .
2.289E+02 2.495E-01 . . * . .
2.395E+02 2.242E-01 . . * . .
2.500E+02 2.038E-01 . . * . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Load voltage peaks at resonant frequency (159.15 Hz)
Just like the low-pass and high-pass
filter designs relying on a series resistance and a parallel "shorting"
component to attenuate unwanted frequencies, this resonant circuit can
never provide full input (source) voltage to the load. That series
resistance will always be dropping some amount of voltage so long as
there is a load resistance connected to the output of the filter.
It should be noted that this form of
band-pass filter circuit is very popular in analog radio tuning
circuitry, for selecting a particular radio frequency from the
multitudes of frequencies available from the antenna. In most analog
radio tuner circuits, the rotating dial for station selection moves a
variable capacitor in a tank circuit.
The variable capacitor and air-core
inductor shown in the above photograph of a simple radio comprise the
main elements in the tank circuit filter used to discriminate one radio
station's signal from another.
Just as we can use series and parallel LC
resonant circuits to pass only those frequencies within a certain range,
we can also use them to block frequencies within a certain range,
creating a band-stop filter. Again, we have two major strategies to
follow in doing this, to use either series or parallel resonance. First,
we'll look at the series variety:
When the series LC combination reaches
resonance, its very low impedance shorts out the signal, dropping it
across resistor R1 and preventing its passage on to the load.
series resonant bandstop filter
v1 1 0 ac 1 sin
r1 1 2 500
l1 2 3 100m
c1 3 0 10u
rload 2 0 1k
.ac lin 20 70 230
.plot ac v(2)
.end
freq v(2) 1.000E-03 1.000E-02 1.000E-01 1.000E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
7.000E+01 3.213E-01 . . . * .
7.842E+01 2.791E-01 . . . * .
8.684E+01 2.401E-01 . . . * .
9.526E+01 2.041E-01 . . . * .
1.037E+02 1.708E-01 . . . * .
1.121E+02 1.399E-01 . . . * .
1.205E+02 1.111E-01 . . .* .
1.289E+02 8.413E-02 . . *. .
1.374E+02 5.887E-02 . . * . .
1.458E+02 3.508E-02 . . * . .
1.542E+02 1.262E-02 . .* . .
1.626E+02 8.644E-03 . *. . .
1.711E+02 2.884E-02 . . * . .
1.795E+02 4.805E-02 . . * . .
1.879E+02 6.638E-02 . . * . .
1.963E+02 8.388E-02 . . *. .
2.047E+02 1.006E-01 . . * .
2.132E+02 1.167E-01 . . .* .
2.216E+02 1.321E-01 . . . * .
2.300E+02 1.469E-01 . . . * .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Notch frequency = LC resonant frequency (159.15 Hz)
Next, we will examine the parallel
resonant band-stop filter:
The parallel LC components present a high
impedance at resonant frequency, thereby blocking the signal from the
load at that frequency. Conversely, it passes signals to the load at any
other frequencies.
parallel resonant bandstop filter
v1 1 0 ac 1 sin
l1 1 2 100m
c1 1 2 10u
rload 2 0 1k
.ac lin 20 100 200
.plot ac v(2)
.end
freq v(2) 3.162E-02 1.000E-01 3.162E-01 1.000E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 9.947E-01 . . . * .
1.053E+02 9.932E-01 . . . * .
1.105E+02 9.911E-01 . . . * .
1.158E+02 9.883E-01 . . . * .
1.211E+02 9.841E-01 . . . * .
1.263E+02 9.778E-01 . . . * .
1.316E+02 9.675E-01 . . . * .
1.368E+02 9.497E-01 . . . *. .
1.421E+02 9.152E-01 . . . *. .
1.474E+02 8.388E-01 . . . * . .
1.526E+02 6.420E-01 . . . * . .
1.579E+02 1.570E-01 . . * . . .
1.632E+02 4.450E-01 . . . * . .
1.684E+02 7.496E-01 . . . * . .
1.737E+02 8.682E-01 . . . * . .
1.789E+02 9.201E-01 . . . *. .
1.842E+02 9.465E-01 . . . *. .
1.895E+02 9.616E-01 . . . * .
1.947E+02 9.710E-01 . . . * .
2.000E+02 9.773E-01 . . . * .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Notch frequency = LC resonant frequency (159.15 Hz)
Once again, notice how the absence of a
series resistor makes for minimum attenuation for all the desired
(passed) signals. The amplitude at the notch frequency, on the other
hand, is very low. In other words, this is a very "selective" filter.
In all these resonant filter designs, the
selectivity depends greatly upon the "purity" of the inductance and
capacitance used. If there is any stray resistance (especially likely in
the inductor), this will diminish the filter's ability to finely
discriminate frequencies, as well as introduce antiresonant effects that
will skew the peak/notch frequency.
A word of caution to those designing
low-pass and high-pass filters is in order at this point. After
assessing the standard RC and LR low-pass and high-pass filter designs,
it might occur to a student that a better, more effective design of
low-pass or high-pass filter might be realized by combining capacitive
and inductive elements together like this:
The inductors should block any high
frequencies, while the capacitor should short out any high frequencies
as well, both working together to allow only low frequency signals to
reach the load.
At first, this seems to be a good
strategy, and eliminates the need for a series resistance. However, the
more insightful student will recognize that any combination of
capacitors and inductors together in a circuit is likely to cause
resonant effects to happen at a certain frequency. Resonance, as we have
seen before, can cause strange things to happen. Let's plot a SPICE
analysis and see what happens over a wide frequency range:
lc lowpass filter
v1 1 0 ac 1 sin
l1 1 2 100m
c1 2 0 1u
l2 2 3 100m
rload 3 0 1k
.ac lin 20 100 1k
.plot ac v(3)
.end
freq v(3) 1.000E-01 3.162E-01 1.000E+00 3.162E+00
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 1.033E+00 . . * .
1.474E+02 1.074E+00 . . .* .
1.947E+02 1.136E+00 . . . * .
2.421E+02 1.228E+00 . . . * .
2.895E+02 1.361E+00 . . . * .
3.368E+02 1.557E+00 . . . * .
3.842E+02 1.853E+00 . . . * .
4.316E+02 2.308E+00 . . . * .
4.789E+02 2.919E+00 . . . *.
5.263E+02 3.185E+00 . . . *
5.737E+02 2.553E+00 . . . * .
6.211E+02 1.802E+00 . . . * .
6.684E+02 1.298E+00 . . . * .
7.158E+02 9.778E-01 . . * .
7.632E+02 7.650E-01 . . * . .
8.105E+02 6.165E-01 . . * . .
8.579E+02 5.084E-01 . . * . .
9.053E+02 4.268E-01 . . * . .
9.526E+02 3.635E-01 . . * . .
1.000E+03 3.133E-01 . * . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
What was supposed to be a low-pass filter
turns out to be a band-pass filter with a peak somewhere around 526 Hz!
The capacitance and inductance in this filter circuit are attaining
resonance at that point, creating a large voltage drop around C1,
which is seen at the load, regardless of L2's attenuating
influence. The output voltage to the load at this point actually exceeds
the input (source) voltage! A little more reflection reveals that if L1
and C2 are at resonance, they will impose a very heavy (very
low impedance) load on the AC source, which might not be good either.
We'll run the same analysis again, only this time plotting C1's
voltage and the source current along with load voltage:
legend:
*: v(3)
+: v(2)
=: i(v1)
freq v(3)
(*)---------- 1.000E-01 3.162E-01 1.000E+00 3.162E+00
(+)---------- 3.162E-01 1.000E+00 3.162E+00 1.000E+01
(=)---------- 1.000E-03 3.162E-03 1.000E-02 3.162E-02
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1.000E+02 1.033E+00 . = + * .
1.474E+02 1.074E+00 . = .+ .* .
1.947E+02 1.136E+00 . = . + . * .
2.421E+02 1.228E+00 . = . + . * .
2.895E+02 1.361E+00 . = . + . * .
3.368E+02 1.557E+00 . .= + . * .
3.842E+02 1.853E+00 . . = + . * .
4.316E+02 2.308E+00 . . = + . * .
4.789E+02 2.919E+00 . . = + *.
5.263E+02 3.185E+00 . . .x *
5.737E+02 2.553E+00 . . +=. * .
6.211E+02 1.802E+00 . . + = . * .
6.684E+02 1.298E+00 . . + = . * .
7.158E+02 9.778E-01 . .+ = * .
7.632E+02 7.650E-01 . + . = * . .
8.105E+02 6.165E-01 . + = * . .
8.579E+02 5.084E-01 . + =. * . .
9.053E+02 4.268E-01 . + = . * . .
9.526E+02 3.635E-01 . + = . * . .
1.000E+03 3.133E-01 . + = * . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Sure enough, we see the voltage across C1
and the source current spiking to a high point at the same frequency
where the load voltage is maximum. If we were expecting this filter to
provide a simple low-pass function, we might be disappointed by the
results.
Despite this unintended resonance,
low-pass filters made up of capacitors and inductors are frequently used
as final stages in AC/DC power supplies to filter the unwanted AC
"ripple" voltage out of the DC converted from AC. Why is this, if this
particular filter design possesses a potentially troublesome resonant
point?
The answer lies in the selection of
filter component sizes and the frequencies encountered from an AC/DC
converter (rectifier). What we're trying to do in an AC/DC power supply
filter is separate DC voltage from a small amount of relatively
high-frequency AC voltage. The filter inductors and capacitors are
generally quite large (several Henrys for the inductors and thousands of
µF for the capacitors is typical), making the filter's resonant
frequency very, very low. DC of course, has a "frequency" of zero, so
there's no way it can make an LC circuit resonate. The ripple voltage,
on the other hand, is a non-sinusoidal AC voltage consisting of a
fundamental frequency at least twice the frequency of the converted AC
voltage, with harmonics many times that in addition. For
plug-in-the-wall power supplies running on 60 Hz AC power (60 Hz United
States; 50 Hz in Europe), the lowest frequency the filter will ever see
is 120 Hz (100 Hz in Europe), which is well above its resonant point.
Therefore, the potentially troublesome resonant point in a such a filter
is completely avoided.
The following SPICE analysis calculates
the voltage output (AC and DC) for such a filter, with series DC and AC
(120 Hz) voltage sources providing a rough approximation of the
mixed-frequency output of an AC/DC converter.
ac/dc power supply filter
v1 1 0 ac 1 sin
v2 2 1 dc
l1 2 3 3
c1 3 0 9500u
l2 3 4 2
rload 4 0 1k
.dc v2 12 12 1
.ac lin 1 120 120
.print dc v(4)
.print ac v(4)
.end
v2 v(4)
1.200E+01 1.200E+01 DC voltage at load = 12 volts
freq v(4)
1.200E+02 3.412E-05 AC voltage at load = 34.12 microvolts
With a full 12 volts DC at the load and
only 34.12 µV of AC left from the 1 volt AC source imposed across the
load, this circuit design proves itself to be a very effective power
supply filter.
The lesson learned here about resonant
effects also applies to the design of high-pass filters using both
capacitors and inductors. So long as the desired and undesired
frequencies are well to either side of the resonant point, the filter
will work okay. But if any signal of significant magnitude close to the
resonant frequency is applied to the input of the filter, strange things
will happen!
- REVIEW:
- Resonant combinations of capacitance
and inductance can be employed to create very effective band-pass and
band-stop filters without the need for added resistance in a circuit
that would diminish the passage of desired frequencies.
-
Summary
As lengthy as this chapter has been up to
this point, it only begins to scratch the surface of filter design. A
quick perusal of any advanced filter design textbook is sufficient to
prove my point. The mathematics involved with component selection and
frequency response prediction is daunting to say the least -- well
beyond the scope of the beginning electronics student. It has been my
intent here to present the basic principles of filter design with as
little math as possible, leaning on the power of the SPICE circuit
analysis program to explore filter performance. The benefit of such
computer simulation software cannot be understated, for the beginning
student or for the working engineer.
Circuit simulation software empowers the
student to explore circuit designs far beyond the reach of their math
skills. With the ability to generate Bode plots and precise figures, an
intuitive understanding of circuit concepts can be attained, which is
something often lost when a student is burdened with the task of solving
lengthy equations by hand. If you are not familiar with the use of SPICE
or other circuit simulation programs, take the time to become so! It
will be of great benefit to your study. To see SPICE analyses presented
in this book is an aid to understanding circuits, but to actually set up
and analyze your own circuit simulations is a much more engaging and
worthwhile endeavor as a student.
Contributors
Contributors to this chapter are listed
in chronological order of their contributions, from most recent to
first.
Jason Starck
(June 2000): HTML document formatting, which led to a much
better-looking second edition.
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