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## Question about tone control frequencies...

Started by Projectile, April 08, 2009, 10:48:29 AM

#### aziltz

#40
Quote from: Projectile on April 14, 2009, 10:35:15 PM
My point was not that the 2nd example is the ONLY way to make a filter.
BUT that's what you said, friend...  I was only reacting to that statement.
Quote from: Projectile on April 14, 2009, 10:35:15 PM
In order have a filter of any kind, you have to create some sort of voltage divider. Like this:

I don't think anyone is saying that R1/C1 make a filter on their own.  HOWEVER!  could it be that R1/C1 have an effect on the R1/R2/C1 filter, and that effect is INDEPENDENT of R2?  Possibly?  Would you sleep at night?  I know I would! LETS LEARN!

Here's some data, I finally figured out how to get 5Spice to work.
Low Pass, R=1k, C=.22uF, like our Mid-Happy Friend the Tube Screamer.  The cursor is centered about -3dB roll-off.

Here's a simulation of Projectile's expertly drawn low-pass/hi-shelf filter (foreshadowing?), using the values, R2=1k, C1 = .22uF and R1 = 220.  It seems to level off at the value of 3.2kHz, as calculated by R1 and C1.  Note: My definition of R2/R1 is reversed, and I forgot to center the cursor on -3dB.

Putting them together, again, cursor is arbitrary, and my schematic definition of R2/R1 is reversed.

note: i've used R2=1 ohm as zero, in order to general two plots together.  There's no difference in the plots between r2=1 ohm and no resistor within this range.

As you can see, the addition of R1 after C levels off the filter, creating a "shelf" in the frequency response.  The overall roll-off of the filter changes slightly, but not enough to warrant shouting BLASPHEMY at those who would explain this using 720Hz and 3.2kHz as "special" frequencies to talk about, more on that in a bit.

Oh, and just for gits and shiggles, here's what happens when we change the upper resistor.  Two curves, R=1K and R=10K.
Note: Cursor is Arbitrary, and my schematic labeling of R1/R2 is reverse from the schematic at the top of the page.

The thing to note here is, the upper leg resistance changes the roll-off frequency, but the point at which things flatten out remains the same, (set by R=220).  To me this says they are independent enough that we can talk about the 220/.22uF on its own.  To clarify, in my professional DIY opinion (joke), it is not incorrect to talk about the 220/.22uF RC constant separately from the 1k/.22uF as they are more or less independent in this configuration.  HIGH FIVE!

Next I intend to simulate how this affects a feedback loop in a gain stage, as well as the complete clip stage and tone stage of the TS, but tonight I've got to study for an Electricity and Magnetism Mid-Term!

Oh, and I'll probably start my own thread, or even post on my website.

Cheers.

If anyone is interested in 5Spice, it seems a bit easier to use than LTSpice, at least for first time users, I can provide some links.  And its FREE.  I can also point to a tutorial, with example for simulations.  It's from the class I teach, associated with my research advisor so i'd rather not post publicly but I will respond to any PM's with the information.

#### Projectile

Now THAT makes sense!!!!!!!!!!

Finally!!!!!!!!!

Thank you! Thank you! Thank you Aziltz!

That is where the 3.2kHz value comes from! It turns out the resistor to ground combination creates another apex of the filter curve where the rolloff "bottoms out". When you put this arrangement in a feedback network, that lower apex becomes the point where the boost becomes level and bass starts rolling off. You can in fact look at it independently. So, You were right about that Aziltz! It's just that that slope of the curve depends upon the relationship of r1 and r2, so if we are going to calculate the technical rolloff point, where the signal is -3db, you would have to look at the relationship of r1 and r2. If you get rid of R2 altogether, then the slope becomes zero and the filter disappears, which is why I was saying that you can't have a filter without R2.  Interesting...  I'm just thinking aloud here.

So, I will eat my words, the AMZ article is basically correct! ...I say "basically" because I still think it's misleading and I don't think the low E string is actually down -20db, since that would depend on the gain pot and the 51K resistor, which set the slope of the curve by defining the other apex...  But I'm not going to nitpick. It's close enough in a general sense, so I'll stand corrected.

The GeofX article however is still appears to be wrong, since it claims very clearly that the signal gets a +6db/octave boost above 3.2KHz. This is not true, since 3.2KHz is actually the point where the boost levels off. Everything above 3.2KHz actually gets the same amount of boost, and since the first filter is rolling off everything above 720Hz at a ratio of -6db per octave, the treble is never fully recovered above 3.2Khz

That matches what I am seeing exactly. It all makes perfect sense now. Awesome! Thank you Aziltz! You are the man!

#### aziltz

Quote from: Projectile on April 15, 2009, 12:38:35 AM
Now THAT makes sense!!!!!!!!!!
Finally!!!!!!!!!
Thank you! Thank you! Thank you Aziltz!

That is where the 3.2kHz value comes from! It turns out the resistor to ground combination creates another apex of the filter curve where the rolloff "bottoms out". When you put this arrangement in a feedback network, that lower apex becomes the point where the boost becomes level and bass starts rolling off. You can in fact look at it independently. So, You were right about that Aziltz! It's just that that slope of the curve depends upon the relationship of r1 and r2, so if we are going to calculate the technical rolloff point, where the signal is -3db, you would have to look at the relationship of r1 and r2. If you get rid of R2 altogether, then the slope becomes zero and the filter disappears, which is why I was saying that you can't have a filter without R2.  Interesting...  I'm just thinking aloud here.

So, I will eat my words, the AMZ article is basically correct! ...I say "basically" because I still think it's misleading and I don't think the low E string is actually down -20db, since that would depend on the gain pot and the 51K resistor, which set the slope of the curve by defining the other apex...  But I'm not going to nitpick. It's close enough in a general sense, so I'll stand corrected.

The GeofX article however is still appears to be wrong, since it claims very clearly that the signal gets a +6db/octave boost above 3.2KHz. This is not true, since 3.2KHz is actually the point where the boost levels off. Everything above 3.2KHz actually gets the same amount of boost, and since the first filter is rolling off everything above 720Hz at a ratio of -6db per octave, the treble is never fully recovered above 3.2Khz

That matches what I am seeing exactly. It all makes perfect sense now. Awesome! Thank you Aziltz! You are the man!

honestly i feel like we've had the same thing in our heads, and it just doesn't translate into words very well.  I wish I had this simulation tool when i first tried to answer your question, we might have avoided some frustration.  This all took about 20 mins to construct and run.  5Spice really is a nice, simple simulation too.

#### Projectile

#43
Quote from: aziltz on April 15, 2009, 12:46:14 AM

honestly i feel like we've had the same thing in our heads, and it just doesn't translate into words very well.  I wish I had this simulation tool when i first tried to answer your question, we might have avoided some frustration.  This all took about 20 mins to construct and run.  5Spice really is a nice, simple simulation too.

Same here.

I wonder how one would actually calculate the "technically correct" -3db rolloff point of this curve, since in this arrangement it's not your typical -6db per octave filter anymore. That's where the AMZ article gets into trouble, although I guess it's close enough for DIY discussion.

I tried messing with LTSpice yesterday, but I wasn't confident enough that I knew what I was doing to post the results. It must have taken me 2 hours just to figure out how to get it to output the plot of a simple RC filter, and then I had to attend to other business. The graphs from 5spice look much nicer though!

Now my next question would be, why does the R2 resistor (using aziltz model) control the point where the filter levels off? And why would that "level off" point be calculated using the same general formula that calculates the roll off point of the R1/C1 combination?  Hmmmmm...

#### Projectile

#44
Duplicate post. Deleted.

#### MohiZ

#45
Wow! This explains everything. Very good results, Aziltz! In a way, we were just all focusing on different points of the response curve.

QuoteThe GeofX article however is still appears to be wrong, since it claims very clearly that the signal gets a +6db/octave boost above 3.2KHz. This is not true, since 3.2KHz is actually the point where the boost levels off. Everything above 3.2KHz actually gets the same amount of boost, and since the first filter is rolling off everything above 720Hz at a ratio of -6db per octave, the treble is never fully recovered above 3.2Khz

Actually, I checked the GEOFEX article, and it says.. quote..
QuoteNote that the "boost" actually just levels off the -6db/octave induced by the 1K/0.22uF network ahead of the active control stage, so the treble is just not being cut any more above the turnover frequency for the tone control stage when fully at "treble".

I think what RG is trying to say here is the same what we have established as of now. However, I think this still can be misleading.

I think the explanation to why the 220 ohm resistor sets the "level-off" frequency is found from the GEOFEX article: "As a series network, at frequencies above the point at where the capacitive impedance is less than 220 ohms (which happens at about 3.2KHz), the network just looks like the 220 ohm resistor." So the high frequency response stops dropping because the combined impedance of the cap and the resistor can't be less than 220 ohms. That means the minimum amplification factor can't be less than 1k/(1k+220R) ~ 0.18 no matter how high the frequency. An amplification factor of 0.18 equals about -15 dB, which seems to be the value that the response levels off to, looking at Aziltz's graphs!

I agree that it's pretty interesting that the way this "level-off" frequency can be calculated with the same formula as a normal single-pole filter. I wonder what's the math behind it. Aziltz, I think it's a good idea to start a new topic about it with a different title, this one is getting a little long.

#### MohiZ

#46
I'm gonna have to answer my own post on this. I'm referring to this

QuoteNow my next question would be, why does the R2 resistor (using aziltz model) control the point where the filter levels off? And why would that "level off" point be calculated using the same general formula that calculates the roll off point of the R1/C1 combination?  Hmmmmm...

I figured this out. Let R1 be the upper resistor 1k, R2 be the lower resistor 220R and C be the cap 0.22uF.

First think about the filter simplified as a normal RC filter: its amplification is calculated as the voltage of a voltage divider, the cap's impedance divided by the sum of the impedances of the cap and resistor. At low frequencies the cap's impedance is large compared to R2, so we can short R2. At high frequencies the cap's impedance is small so we can short the cap.

Now we calculate the amplification factors of both of these simplifications.

Next we calculate the frequency where the amplification of these simplified models is equal, that is, when the frequency response levels off.

And the answer is, oddly enough, the same as the roll off point of the R2/C combination would be. Only one question: why this isn't properly explained in these articles? I hope you can make out the hand-written equations

In conclusion here are the formulas for this type of filters, with the example values of our beloved TS circuit  :
1) Initial -3dB roll-off: f = 1/(2*pi*R1*C) ~ 720 Hz
2) Level-off frequency: f = 1/(2*pi*R2*C) ~ 3.3 kHz
3) Level-off amplification: R1/(R1+R2). In decibels this is 20*log [the answer] ~ -15 dB.
So at first there's no attenuation, then at 720 Hz the volume starts to drop, and at 3.3 kHz the volume has dropped to -15 dB and ceases to drop, staying at that level even at higher frequencies. All of this is backed by the graphs of aziltz, so the approximation works extremely well!

#### earthtonesaudio

Quote from: Projectile on April 13, 2009, 07:35:26 PM
Look, you can't build an RC filter like this:

I know I'm splitting hairs here, but that's incorrect.  Real-life op-amps have low, but non-zero output resistance.  So while you're correct that you can't build an RC filter without an "R," you will never find an op-amp without output resistance, so in practice, you can build a filter like that.

Carry on...

#### aziltz

MohiZ,

That's Exactly the kind of stuff I got when doing the math yesterday, i just had trouble making those approximations so I went straight for the simulations, but I got the same curves graphing the complex impedance-calculated gain on my TI89 as i did with the simulation.

So at high frequencies, the lower resistor causes the filter to act like a voltage divider, genius!

#### Projectile

#49
QuoteNote that the "boost" actually just levels off the -6db/octave induced by the 1K/0.22uF network ahead of the active control stage, so the treble is just not being cut any more above the turnover frequency for the tone control stage when fully at "treble".

Still seems wrong to me.

The treble IS being cut ABOVE the 3.2KHz. It is compensated for, or rather "brought back up to level", BELOW the 3.2KHZ turnover. The GeoFX article has it backwards.

He did get the first part right though, so he must understand what is going on.  It seems like he just made a simple mistake. Anyway, I sent a PM to R.G. about this, so hopefully he can clear this up himself.

Quote from: MohiZ on April 15, 2009, 07:18:41 AM
I agree that it's pretty interesting that the way this "level-off" frequency can be calculated with the same formula as a normal single-pole filter. I wonder what's the math behind it. Aziltz, I think it's a good idea to start a new topic about it with a different title, this one is getting a little long.

That's what's still baffling me.

I noticed that the 220ohm resistor was creating a "foor" that no frequency could fall below, and I could see that it was creating a second apex in the curve, but there was no way in a million years I would have guessed that you could plug the 220 ohm resistor into the same formula to determine that lower apex. It just doesn't make any sense to me, since that resistor is performing an entirely different function in the circuit. Weird.

Actually I may have an explanation, but its a bit wacky, and since I've been having trouble communicating my other points in this thread, I don't know even I should even try. I'm probably digging a hole with this one. My mathematics terms are a bit sketchy at this point in my life, but here goes:

EDIT: Mohiz came to a similar conclusion while I was typing this, but I'm going to post my analysis also since I had already finished typing it before I saw his...

Let's call the 1K resistor "R1" and the 220 ohm resistor "R2" and the capacitor "C1", like in Azilt's graph.

Ignore R2 for a moment.

R1 and C1 form voltage divider that varies with frequency according to the value of C1. Together, they can be used to calculate the rolloff point off the filter.  They are inversely related, in that, if you double R1 you must half C1 in order to get the same rolloff point, and vice versa.

Now, let's ignore C1 for a moment and look at the relationship between the two resistors. R1 and R2 form a voltage divider with a set value. They can be used to calculate the level which the signal cannot fall below. C1 only controls frequencies above this "floor" by adding to the bottom leg of the voltage divider that is initially set by R2.  R1 and R2 are directly related, in that if you double R1 you must double R2 in order to get the same frequency "floor", and vice versa.

Where the rolloff curve set by R1 and C1 meets the "floor", created by R1 and R2, is a second apex, or the "leveling off" point. I imagine that there is some formula you could use that involves all three variables to calculate that apex, and it would seem like it would be necessary. But, it's not necessary, and here's the trick...

Since R1 and C1 are inversely related in the same way that R1 and R2 are directly related, you can essentially plug R2 and C1 into the same formula that describes the relationship between R1 and C1 to get a answer that describes that lower apex independent of R1. It works because all 3 terms share the same mahtematical relationship in the same voltage divider. This explains why you can manipulate R2 and C1 independently of R1, even though R1 is essential in creating the filter.

Say you half the value of R2. That means that the relationship between R2 and C1 doubles and the frequency of the lower apex increases. But that also means that the value of voltage divider formed by R1 and R2 decreases.  So, whatever value you plug in for R1, the lower apex is going to be the same.

It's just confusing as hell because when you are just thinking in terms of electrons in a wire there should be no reason why R2 controls the frequency of anything without R1! You can't even calculate a voltage divider without R1 ...but the math works because the relationship of the terms allows for that simple substitution.  R1 and C2 sets the initial rolloff point of the filter and the relationship between R1 and R2 sets the slope of the curve. If you take away R1, there is no curve, and no filter, but that doesn't mean you can't calculate the apex of the lower part of the curve intependant of R1. We just can't talk about it as the "technical" -3db rolloff point anymore, because it's not. It just describes the general location of the apex.

At least that makes some sense to me.

#### Projectile

Quote from: MohiZ on April 15, 2009, 08:08:29 AM
Only one question: why this isn't properly explained in these articles? I hope you can make out the hand-written equations

Yes, really!

It's all so clear now. I've seen these particular implementations of filters all over the place, but no adequate explanations. I scoured the internet during the course of this thread and could only find either basic descriptions of RC filters, or complex mathematics that went way over my head. I'm really surprised this has never been discussed here before. It seems pretty important to understanding many of these stompbox circuits. I'm really glad we finally all came to the same conclusions. This stuff is fascinating!

This thread has actually encouraged me to go back to college and take some electronics courses. Maybe I'll stop being a lazy musician and make something of my life after all.

Thanks to everyone who contributed to solving this little puzzle.

#### aziltz

i guess the thing about calculating a time-constant or a cut-off frequency from R&C , is it all has to do with when the Cap in question has an equal impedance with another element of the nearby circuit, so R1 or R2.

What I'm trying to suggest here is, at one frequency, the impedance of C is equal to R1, and at another frequency its equal to R2, and I believe this works for more complicated filters in general.  Thoughts?

#### Projectile

Quote from: aziltz on April 15, 2009, 09:56:05 AM
i guess the thing about calculating a time-constant or a cut-off frequency from R&C , is it all has to do with when the Cap in question has an equal impedance with another element of the nearby circuit, so R1 or R2.

What I'm trying to suggest here is, at one frequency, the impedance of C is equal to R1, and at another frequency its equal to R2, and I believe this works for more complicated filters in general.  Thoughts?

I follow you on the general idea, but you can you give an example of a more complicated circuit where you think the RC calculation would work like this? I'm just not so sure this little trick substitution that works so perfectly with r2 and c1 would apply in all situations.

#### Projectile

Quote from: earthtonesaudio on April 15, 2009, 08:42:00 AM
Quote from: Projectile on April 13, 2009, 07:35:26 PM
Look, you can't build an RC filter like this:

I know I'm splitting hairs here, but that's incorrect.  Real-life op-amps have low, but non-zero output resistance.  So while you're correct that you can't build an RC filter without an "R," you will never find an op-amp without output resistance, so in practice, you can build a filter like that.

Carry on...

I'm aware of that. Funny, I originally had a tangent where I explained that the output impedance of an opamp could act as R2, but I omitted it before posting because I thought it just muddled the issue and distracted from the general point I was trying to make. The graphic doesn't specify that it is in the feedback loop of an opamp anyway, and when I did describe that circuit in the feedback loop of an opamp I think I mentioned that the opamp had a low output impedance just so that nobody would call me on the fact that an opamp's output in fact has an impedance.

If we are going to really split hairs that much, then I would still be calling the AMZ article incorrect, but I'm just assuming that certain things were omitted for the sake of brevity.

#### aziltz

Quote from: Projectile on April 15, 2009, 10:13:56 AM
I follow you on the general idea, but you can you give an example of a more complicated circuit where you think the RC calculation would work like this? I'm just not so sure this little trick substitution that works so perfectly with r2 and c1 would apply in all situations.
like our example above, i think since 720 and 3.2k are far enough apart, you can describe things separately.

i'm not sure its universal, and I won't make a claim to that.  it just appears that at high/low frequency approximations the gain can definitely be simplified, as MohiZ showed.  i just wanted to suggest that there's something special happening at each combination of C and Rx (1 or 2).  I think the R2/C cut-off (3.2kHz) relates to the High Frequency limit MohiZ took.  Can anyone flesh that out?

other filter arrangements to try would be, R1+C as the upper leg and R2 as the lower leg (High Pass + Low Shelf?)

or R1||C as one leg and R2 as the other leg, in either a Low or High Pass arrangement.

If i get some time tonight I'll simulate these.

#### MohiZ

It's amazing to think that we thought about the same thing at the same time! This is great stuff, I wouldn't have even thought about it without this topic. Now we can create all kinds of interesting filters instead of just the usual low-pass or high-pass.

The key definitely lies in simplifyin these things. Think about how much we had to simplify this tone stage for instance; first we took apart the tone control's relation to the op-amp and even after that the tone control itself had to be broken to pieces before we gained full understanding about this. I agree that the approximations work so well because the two corner frequencies are far enough apart.. Otherwise there would be more cross-talk. Aziltz's view that something interesting happens at the points when the impedance of the cap equals one of the resistors' impedances makes sense. The cap's impedance is the only one that changes due to frequency, and it changes in a non-linear way too.

#### MikeH

#56
Nevermind...
"Sounds like a Fab Metal to me." -DougH

#### biggy boy

Quote from: MikeH on April 15, 2009, 12:41:57 PM
Nevermind...
Come on now don't keep any secrets!

#### MikeH

Quote from: biggy boy on April 15, 2009, 01:02:23 PM
Quote from: MikeH on April 15, 2009, 12:41:57 PM
Nevermind...
Come on now don't keep any secrets!

Post removed due to excess snarkieness.
"Sounds like a Fab Metal to me." -DougH

#59