64.36% Low Pass Filters

[WGTP]

You guessed it.  That is how much the -3db or rolloff point drops when 2 identical first order (cap to ground for example) low pass filters are cascaded (theoretically).  Following is a nontechnical explanation by a simple minded guitar player. 

Some of the ROG sims/ems have 2 resistors and 2 capacitors of the same value at the end of the circuit to roll off high frequencies.  This creates a 2nd order or 12db/octave roll off.  A single resistor/cap would be a 1st order filter at 6db/octave.  According to the formula I found (but don't really understand) the -3db point will drop to 64.36% of the value of the single filter and the rate increase to 12db/octave.

For example a 10K/.001uf low pass filter (like at the end of the ever popular Distortion+, discounting the effect of the output pot) would result in a 16,000Hz roll off point with a 6db/oct. rate.  Cascading 2 would result in a 10,300Hz roll off point with a 12db/oct. rate.

Using a .002uf cap would produce more fizz reducing results with a single filter yielding 8KHz and a double yielding 5.15KHz. 

Often you will see low pass filters scattered through out a circuit.  I ASSUME these compound, but distortion will provide additional harmonics as the signal moves thru the circuit.

I have seen at least 4 different locations in a Fuzz Face where designers add small value caps (particularly in SI FF) to reduce the treble content.  Again note the FF type circuits at ROG.  Another would be 2 op amp stages cascaded with a cap in parallel with the resistor in the feed back loop.

My simple trial and error method involves cranking the treble up on the amp and trying different value caps at one location until I hear a slight reduction in hiss then moving to a different location/stage in the circuit and adding another cap I can slightly hear.  This then SHOULD result in an approximation of a 12/db/oct filter. 

Remember different resistances will produce different roll off points as well and the actual impedance at a certain location will probably be different than our simple calculations. 

Certainly if this info is erroneous, someone will need to speak up.

Hope this helps with the fizzes.   

[John Lyons]

Good info to keep in mind. Thanks

John

[George Giblet]

>That is how much the -3db or rolloff point drops when 2 identical first order (cap to ground for example) low pass filters are cascaded (theoretically).

That result is when there is a buffer in between the two RC stages.  The easy way to understand it is the -3dB point of the whole filter is the -1.5dB point of one of the RC filters.

The case where the RC networks are connected together without a buffer in between changes things because the filters interract with one another.   If you make the second R much larger and reduce the second cap by the same factor (so the RC product remains the same) then you can approximate the buffered case.  You need the second R to be 5 to 10 times the first to get a reasonable approximation.  As the second R gets smaller not only does the -3dB change but the roll-off gets less steep.

                          Cut-off
First order             1
Cascade                0.65
Second R x10        0.60
Second R x5          0.56           ; about 10% error at 5 times
Second R x2          0.47
Second R x1          0.37

[WGTP]

Thanks for the additional info.  I suspected as much, but didn't know for sure.  I tried to indicate the 2 filters should be at different locations, that is my attempt at explaining electronically isolated or buffered.

I started thinking the same is probably true for high pass filters as well, and by dividing 1 by .6436 obtained 1.55 which MAY be the value to use on cascaded high pass flters???   

[George Giblet]

>by dividing 1 by .6436 obtained 1.55
Yep, the idea works for High Pass.

(Not just my idea but a well known filter result.  HPFs are often designed by starting with an LPF, usually under the topic of frequency transformations and dual circuits).