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.
-
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