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.
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