# Multiple Feedback Band-Pass filter

Band-pass filters are filters that are designed to let only a certain set of frequencies pass through. Overtime a myriad of different topologies have been developed by some very creative designers. Passive vs active, RC-filters vs LC filters, combinations of low pass and high pass filters, with gain and without gain, etc. One of the topologies that are often used for CW filters in receivers is the multiple feedback band-pass filter :

Band-pass filters are filters that are designed to use only a certain set of frequencies pass through. If you want to describe the filter function, you need only a couple of parameters:

• Center frequency fo = the center of the pass band eq the peak of the response curve
• Bandwidth BW = fH – fL = the upper and lower frequencies that are passed through. These point are defined where the gain has dropped 0.707 of the maximum gain or the -3dB point.
• Quality factor Q = f0 / BW = tells how the width of the passband relates to the center frequency. If the Q is high, the passband is relatively small and vice versa.
• Gain G = the gain or signal amplification a frequency f0.

In the Multiple Feedback Band-pass topology all these characteristics can be set individually. Therefor this topology is a favorite with many designers.

If you want to dimension the value of the component of the filter, you need to make decision on your requirements of the above parameters. (No panic, if you are not 100% sure, after this you are going to build the filter anyhow and you can always change some values to components to tweak the filter).
Choose a value for C1 = C2. If not sure, start with a value of 10nF. If the resistor value comes out too high or too low, these can always be changed.
Then calculate, R1, R2 and R3 using the following formula :

$R_1 =\frac {Q}{G 2 \pi f_0 C}$
$R_2 =\frac {Q} { (2 Q^2 -G) 2 \pi f_0 C }$
$R_3 =\frac {2 Q} {2 \pi f_0 C}$

Or if you assume k = 2 π f0 C :

$R_1 = \frac {Q}{G k}$
$R_2 = \frac{Q}{(2Q^2-G)k}$
$R_3 = \frac {2Q}{k}$

If you want to reverse engineer an existing filter, to understand it’s parameters, use the following formula:

$f_0 = {{1} \over {2 \pi C} } \sqrt { \frac {R_1 + R_2}{R_1 R_2 R_3} }$
$G = – \frac {R_3}{2 R_1}$
$Q = \pi R_3 C f_0$
$BW = \frac {1}{\pi R_3 C} = \frac {f_0}{Q}$

If you want to understand and see the derivation of the above of formulas, I can highly recommend ‘Halbleiter-Schaltungstechnik‘ from U. Tietze & Ch. Schenk. In chapter 13.7.2. (seite 308-310) they explain it all. Luckily for us, it is not necessary to understand how to formulas came to be, before we can use them!

## Literature

The Multiple Feedback Band-Pass filter topology is used by many professional and amateur radio designers. An evening browsing through my library yielded the following the radio designs where this filter topology is used (listed in random order) :

1. James L. Tonne W4ENE “An Improved Audio-Frequency Bandpass Filter for Morse Code Reception”, QEX March/April 2017, page 31ff
2. PY2OHH, ACTIVE CW BAND PASS FILTER
3. PA3HDF, Jakob Schripsema, Direct Conversion receiver for 80m and 40m
4. K4ICY, Home Brew CW Filter
5. Blair Babida, Multiple-Feedback Band-Pass Filter
6. W1FB, Doug DeMaw, A Switched Variable Selectivity AF Filter, W1FB QRP Notebook, page 67-68. Also published in W1FB design Notebook, page 133-134 as ‘Four-Pole RC Active CW Filter’
7. W1FB Doug DeMaw, ‘Two-Pole RC Active Audio Filter’, W1FB design Notebook, page 60-61. Alos published in Solid State Design for the radio amateur. Wes Hayward W7ZOI and Doug DeMaw W1FB, page 138.
8. Eric T. Red, ‘KW-Monobander für 15/50/80m mit 9 MHz Zf’, Funkempfänger-SchaltungsTechnik praxisorientiert, seite 121
9. Eric T. Red, ‘KW-DC_Monobander für 15/17/20 m’, Funkempfänger-SchaltungsTechnik praxisorientiert, seite 117. This is a tunable filter
10. Eric T. Red, ‘2-stufiges aktives Nf-CW-Filter’, Funkempfänger-SchaltungsTechnik praxisorientiert, seite 95.
11. Eric T. Red, ‘Nf-AGC-Zug mit aktivem CW-Filter’, Funkempfänger-SchaltungsTechnik praxisorientiert, seite 91.
12. Eric T. Red, RX-CW Filters, Arbeitsbuch für den HF-Techniker, seite 106.
13. Ham Radio Today, Audio Filter for CW, January 1987, page 49-50
14. DX-394, source: Boat Anchor Manual Archive.
15. GM3OXX, An Active Filter, GQRP Circuit Handbook 1983, page 57

Looking at the table above, the center frequency of most of the fixed filters is somewhere in the range of 700 Hz – 900 Hz. Only the first filter, designed by OM James L. Tonne W3ENE, is centered around 500Hz. Most filters do have a gain of somewhere between 1.0 and 1.3. There are three notable exceptions. The Realistic DX-394 has a modest gain of 4.1x. PA3HDF in his 40m DC-receiver uses a 2 stage cascaded filter with the gain for each stage set at 9x for a total gain of 38 dB. Eric T. Red deviates from this patterns with designs with gains of up to 20x per stage. OM Red has done some other interesting stuff in some of his design. Filter #9 and #10 are made tunable. The center frequency fc can be varied from circa 500 Hz to 1500 Hz. To achieve this only R2 needs to be changed. This not only raises the center frequency but also raises the Q proportionally, so that the bandwidth remains the same!
Filter #11 and filter #12 use two cascaded filters in which the center frequency fc has been offset a little bit. In this way, the pass band is widened a little bit while at the same time using the steep skirts and rapid roll of associated with filters with high Qs.

All really interesting stuff! I would love to include some circuits of the designs of Eric T. Red but those probably still have copyright … so it’s not fair to throw them out on the Internet…

I ran some simulations in LTSpice of the two stage tunable filter from #9, however. You can see the circuit below. (the files can be downloaded from the downloads section).

And below the output of the filter for several settings of R2. The value of the potentiometer is changed in 20% increments (thus 94 Ohms for a 470 Oms value). It can be seen that the frequency does not change linearly with the settings of R2. In practice this will not be a real problem, tuning will almost always be done by ear. It can also be seen that width of the peak does (almost) change with the setting of R2 => thus the filter bandwidth remains unchanged. Jus as predicted! The peak output of the filter is at circa +5 dB, the input signal is 10mV (-40 dB). The total gain is thus +45 dB. This nicely aligns with the theoretical value of : 20 x log (13.52) = 45.2 dB. Another noteworthy thing; the initial fall off near the center frequency fc is quit sharp. For frequencies further away from this value, the attenuation slopes becomes less steep, however. It might therefor be beneficial to complement this filter with a low pass filter with a corner frequency slightly above the highest filter setting. For more information see the article from OM James L. Tonne W4ENE in the literature above.

## Tips

• Keep the value of C reasonable : somewhere between 1nF and 100nF, use 10nF as a starting point
• Make sure that the OpAmp has sufficient gain-bandwidth product. If designing a filter with a fc of 1khz and a gain of 10, make sure the gain bandwidth is at least 100x higher eq. 1 MHz.
• Use a well-filtered power supply. It won’t do, to filter out noise and/or unwanted CW signals only to introduce hum
• Use 1% metal-film resistors for low noise and good matching
• It is better to cascade multiple filters than to design a single filter with very high gain or Q.
• If cascading filters, select best matching capacitors from a larger batch. The absolute value is often less important (only shifts fc) than the absolute value.