Hi All --
Here is some info on active filters, especially Bessel:
Bessel filters are preferred for seismic and other data-gathering work
where the signals come in at various frequencies and where time correlation
of the data is important. Bessel filters have linear-phase response, which
means the phase shift varies approximately linearly with frequency. Since
phase shift through the filter varies linearly with frequency, phase delay
and group delay are constant. Other filter types don't maintain this time
correlation and can distort the waveform since the various frequency
components get delayed by differing amounts. This is bad for something
like seismic waveforms where the shape of the waveform is of major importance.
A note about cutoff frequency... Two ways to define cutoff frequency are
in common use: The -3db point, and the intersection of asymptotes. The
former is really the point where the amplitude attenuation through the
filter is the square root of two, and the latter is the intersection of the
tangents to the response curve at zero freqeuncy and infinite frequency.
As I remember, these two are the same for Butterworth filters, but differ
for other types. Try the filter mathematically before buying parts.
Over the years I have saved various articles on filters from electronic
journals. The following are some pertinent to the topic, in no particular
order:
1. Kincaid, Russel, "RC Filter Design by the Numbers" The Electronic
Engineer Oct 1968. p57-63
2. Al-Nasser, Farouk "Tables Speed Design of Low-Pass Active Filters" EDN
Magazine, March 15, 1971. p23-32
3. Al-Nasser, Farouk "Tables Shorten Design Time for Active Filters"
Electronics Magazine October 23, 1972 p113-118
4. Shepard, Robert R., "Active Filters: part 12 Short Cuts to Network
Design" Electronics Magazine August 18, 1969 p82-91
5. Warren, George H., "Computer-aided-design Program Supplies
Low-pass-filter Data" EDN Magazine August 20, 1980. p148-150
6. Keith, Timothy, "Design Active Low-Pass Filters" Electronic Design
Magazine September 1, 1977 p144-145
7. Geffe, Philip R. "How to Build High-quality Filters out of Low-Quality
Parts" Electronics Magazine November 11, 1976 p111-
For notch filters:
8. Darilek, Glenn and Tranbarger, Oren "Try a Wien-Bridge Network Instead
of a Twin-T..." Electronic Design Magazine February 1, 1978. p80-81
In addition, there is a good chart comparing the characteristics of
Butterworth, Transitional Butterworth-Thompson, and Bessel filters in
Electronic Design Magazine April 12, 1967 p81. I don't have the article
reference.
Reference 4 is probably the best for cookbook-type Bessel filter
construction. References 2 is also good, but the capacitor values in the
tables contain some errors. When designing a filter, it's best to check
the response mathematically if you can before you build the filter.
These articles are written around filters built using second- and
third-order stages, cascaded if necessary for higher-order filters. Each
stage uses a unity-gain amplifier (Shown as a box below) to isolate the
filter output from its load and from the feedback component. The
unity-gain amplifier is usually an op-amp with its output connected to its
inverting input. The reference designators for the capacitors differ in
the various articles, so be sure you understand what the author means.
The second-order stage is this:
C1
+-------------------||--------+
| |
| +------+ |
R1 | R2 | Amp | |
Input O---\/\/\/\--+--\/\/\/\--+----| X1 |-----+------0 Output
| | |
--- +------+
C2 ---
|
|
---
\ /
V
The third-order stage is this:
C2
+-------------------||--------+
| |
| +------+ |
R1 R2 | R3 | Amp | |
Input O---\/\/\/\---+--\/\/\/\--+--\/\/\/\--+----| X1 |-----+------0 Output
| | | |
--- --- +------+
C1 --- C3 ---
| |
| |
--- ---
\ / \ /
V V
For modeling purposes, the following are freqeuncy response equations for
the second- and third-order stages above. Note that s is a complex value
and must be treated as such. Once the real and imaginary components of the
equations are are determined, amplitude response and phase shift are easily
computed.
The pitfalls I have found with modeling filters are 1) make sure references
to the capacitors are the same (they are easy to get mixed up), 2) make
sure of the frequency and resistor scaling, and 3) make sure frequency is
multiplied by 2*pi.
The frequency response of the second-order stage is given by:
Eo/Ei = 1/(A*s^2 + B*s + 1)
where
A = R1*R2*C1*C2
B = C2*(R1 + R2)
s = sqrt(-1) * 2 * pi * frequency
The frequency response of the third-order stage is given by:
Eo/Ei = 1/(A*s^3 + B*s^2 + C*s + 1)
where
A = R1*R2*R3*C1*C2*C3
B = C2*C3*R2*R3 + C2*C3*R1*R3 + C1*C3*R1*R3 + C1*C3*R1*R2
C = R3*C3 + R2*C3 + R1*C3 + R1*C1
s = sqrt(-1) * 2 * pi * frequency
Now for the data:
The following are tables of capacitor values in Farads for a 1Hz Bessel
filter of various orders. Each filter is constructed from combinations of
cascaded 2- and 3-pole sections.
The values scale linearly with frequency and resistor value. To increase
cutoff freqeuncy, reduce the capacitor values. To increase resistor
values, reduce capacitor values.
First order filters are trivial ones. These are single R-C's.
C1 = 0.1591
2nd Order
C1 = 0.1442 C2 = 0.1082
3rd Order
C1 = 0.1572 C2 = 0.2264 C3 = 0.0403
4th Order
C1 = 0.1169 C2 = 0.1073
C1 = 0.1610 C2 = 0.0620
5th Order
C1 = 0.1656 C2 = 0.0493
C1 = 0.1386 C2 = 0.1605 C3 = 0.0492
6th Order
C1 = 0.1010 C2 = 0.0970
C1 = 0.1149 C2 = 0.0769
C1 = 0.1707 C2 = 0.0407
7th Order
C1 = 0.1153 C2 = 0.0660
C1 = 0.1747 C2 = 0.0344
C1 = 0.1240 C2 = 0.1357 C3 = 0.0481
8th Order
C1 = 0.0902 C2 = 0.0881
C1 = 0.0969 C2 = 0.0773
C1 = 0.1154 C2 = 0.0571
C1 = 0.1776 C2 = 0.0295
9th Order
C1 = 0.0962 C2 = 0.0692
C1 = 0.1162 C2 = 0.0502
C1 = 0.1809 C2 = 0.0259
C1 = 0.1125 C2 = 0.1203 C3 = 0.0453
10th Order
C1 = 0.0823 C2 = 0.0810
C1 = 0.0861 C2 = 0.0745
C1 = 0.0954 C2 = 0.0620
C1 = 0.1165 C2 = 0.0444
C1 = 0.1831 C2 = 0.0228
-- Karl
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Larry Cochrane <cochrane@..............>