From: Karl Cunningham karlc@.......

Date: Tue, 30 May 2000 14:14:28 -0700

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 __________________________________________________________ Public Seismic Network Mailing List (PSN-L)

Larry Cochrane <cochrane@..............>