Help Me Understand Gain of Opamp Circuit (Blues Breaker)

[clubrulz]

I have only recently decided to build my own effects pedals. I started with a bag of parts from some friends who have been tweaking pedals and amps for years. I decided to go for the ProCo Rat Clone and a Marshall Blues Breaker based on the BYOC British Blues pedal. I have not purchased the kits but rather just used the schematics to bread board the circuit and absolutely love the RAT. I am now playing around with the Blues Breaker and would like to get a little more gain. I know I can blindly follow the schematic and simply replace the 220 k resistor in the second stage (inverting gain) with a higher value and get more gain. The Engineer in me needs to do the math to understand why this works.
I have read Wamplers, books as well as The Stompbox Cookbook, and numerous others regarding OpAmps. So here are my questions:
In the first stage (non-inverting) the schematic shows a 100K pot as the feedback resistor with a 3.3k – 0.01uf cap and 4.7K - 0.01uf cap in parallel to ground. If I understand correctly the gain of this stage would be 1/R1 = 1/3.3+1/4.7 = 1.9k; so with Rf = 0-100 max gain would be 100/1.9 = 52.63
In the inverting stage, my understanding is that the 1/Rf = 1/220+1/6.8 = 6.6k while the R1 = 10k
Gain is -6.6/10 = -0.66
Mod documents say to simply change the 220k to a larger value, but because it is in parallel to the 6.8k it doesn't really change the effective value to any great degree. Am I missing something?
Would it not be easier to change the 3.3k and 4.7k in stage 1 to a lower value to increase the gain in stage 1? I would also change the caps as I believe they are also acting as frequency filters.
How is overall gain calculated Stage 1 x Stage 2? So 52.63*-0.66 = -34.73??

[Mark Hammer]

Do you want more output level, gain, or distortion?  People often confuse these three.

A circuit can have lots of gain applied, and gobs of distortion, but not much output level.  It can have some distortion and produce lots of output, without necessarily applying all that much gain.  It can also apply gain with lots of output, but without any distortion.

The second stage in the circuit - a mild clipping stage - has a max gain of 22x.  Because the one pot simultaneously increases the gain for BOTH stage 1 and 2, when stage 2 is set to 22x, stage 1 is also set to a gain of roughly 22x.  And since gain is multiplicative, the joint gain is 22x22=490...not too shabby.  And also capable of substantial output.

If you want more output, you can simply add another diode in series with D1/D2 and D3/D4, or simply replace them with a back-to-back pair of red LEDs.

That will raise the clipping threshold, though, so you won't get as much clipping, even though you'll have more output.

If you want more gain and some additional distortion, and aren't concerned with output level, you can up the gain of stage 2 by making R7 a higher value, like 330k.  That will score you a max gain from the overall circuit of around 725x.  There will likely be some increased level, but by keeping the threshold fixed it will show up as louder soft passages, with little real change to the loudest parts.

If you simply want more distortion, just drop R6 entirely, or consider replacing it with a 10k variable resistance, either in panel-mount pot, or trimpot form.  R6 "softens" the clipping effects of the diodes.  Increasing that resistance lessens their clipping impact, and reducing the resistance increases their clipping "hardness".

[clubrulz]

Thanks for the quick response. I see where you are coming from stage 1 = 100/4.7 and stage 2 = -220/100, but cannot understand why R4 doesn't come into the equation for the gain in stage 1 or why R6 and R5 do not come into the equation for the gain in stage 2.

While we are looking at this, what is the reason for going with a non-inverting stage followed by an inverting stage?

[nocentelli]

http://www.diystompboxes.com/smfforum/index.php?topic=104029.0

The pot affects the gain of both stages.

[Mark Hammer]

Thanks for the quick response. I see where you are coming from stage 1 = 100/4.7 and stage 2 = -220/100, but cannot understand why R4 doesn't come into the equation for the gain in stage 1 or why R6 and R5 do not come into the equation for the gain in stage 2.

Well, you're right.  It does.  But R4 affects only mids and highs, rather than the full bandwidth gain, and I was trying to keep it simple.  Still, good eye!

The input resistance of stage 2 is made up of the sum of R5 and the sum of whatever portion of the pot is not in the feedback loop of stage 1.  As for R6, keep in mind that when the forward voltage is reached, D1-D4 conduct as if they are essentially straight wire.  With R6 in series, the diodes behave like straight-wire-plus-6k8, such that it is not quite as "hard" a clip.  Remember that when the diodes conduct and feedback passes through R6, that path results in less gain than if the signal has to pass through R7 only.

When the diodes conduct, that results in a sudden drop in gain which, in turn, results in there being insufficient signal amplitude to make the diodes continue conducting.  Since a minimum feedback resistance of 6k8 still provides some gain, it becomes easier for the signal to make the diodes continue to conduct.  BUt since the gain is modest, there is not enough drop in feedback level to make R7 become the primary feedback path.  Make sense?

While we are looking at this, what is the reason for going with a non-inverting stage followed by an inverting stage?

The use of a non-inverting first stage is, I gather, to take advantage of its somewhat higher input impedance, and the opportunities for tone shaping via the ground leg from the inverting pin.

The second stage is inverting because: a) you don't need to worry about its input impedance, and b) the gain of such a stage is set by the input and/or feedback resistance.  The gain pot gets to pull double duty as both feedback resistance for stage 1 and input resistance for stage 2, in reciprocal fashion.  Happily, that reciprocal action allows the single pot to adjust gain in the same direction for both stages.  And by using multiplicative gain, rather than sticking all the gain in one stage, the chip behaves better.

Here's an idea.  I shrunk the BYOC drawing down a bit and inserted a few new parts.  R7 has been dropped to 150k.  That gives the stage a max gain of 15x, instead of 22x, making a total circuit gain of 22x15=330, rather than 490.  So, still hot, but not quite as much.  R7 is augmented by part of a 100k pot.  When the wiper is all the way at one end, that 100k is added to the 150k, making the feedback resistance 250k, but just as importantly no resistance in series with the diodes.  So, set the gain high and you get hard clipping.  As you rotate the new pot in the other direction, though, you subtract feedback resistance, and add series resistance with the diodes to restore the softer clipping.  The new R6 sets a limit to how much series resistance.  I picked 8k2, because if you have the new pot set so that there is 70k on one side of its wiper, and 30k on the diode side, 70k+150k gets you 220k, and 30k in parallel with 8k2 gets you 6k5...close enough to the original specs.  Rotated such that the full100k is in parallel with 8k2, that gets us 7k6 in series with the diodes.  That, in tandem with less gain in that stage will get us softer distortion.

A simple, but interesting twist on a classic.  Might be worth a try.

[Gus]

Are the BYOC schematics correct?  If so C5 is the wrong value.  IIRC the stock circuit had two .22uf caps and two 4.7k resistors  in between stages used with the original switching.  
So C5 should be .1uf with 10k for the same RC AND why a 1meg input antipop? to take away even more high end from the guitar?  That antipop can be a higher value with a jfet input opamp.

Increasing the 2nd stage 220k feedback resistor does not make sense to me all that does in change the balance of how the gain changes between stages before the two diodes conduct with the 6.8K in series and limit the ouput.

The nice thing about the Bluesbreaker is how the gain changes in both stages with the use of one volume control.  IMO the designer(s) at Marshall did a nice job with how both stages gain changes.  
Also the diode limiting clipping is done with an inverting amp stage.  The first stage is non inverting allowing higher input resistance and some EQ in the - to ground network.

IMO this petal is good sounding stock

[Mark Hammer]

The Marshall original used a rather different bypassing arrangement, and grounded a point between stage 1 and 2 in bypass mode, presumably to avoid potentially applying a gain of 490x to noise and having it leak into the audio path.  The BYOC version is predicated on more standard bypassing with a switching action at the output and input.  hence the difference in components between stage 1 and 2.

At least that's what I can surmise from the diagram.

[clubrulz]

Are the BYOC schematics correct?  If so C5 is the wrong value.  IIRC the stock circuit had two .22uf caps and two 4.7k resistors  in between stages used with the original switching.  
So C5 should be .1uf with 10k for the same RC AND why a 1meg input antipop? to take away even more high end from the guitar?  That antipop can be a higher value with a jfet input opamp.

You are correct, the original had a .22uf cap followed by a 4.7k resistor and then a second .22uf cap followed by a 4.7k resistor. I assumed the caps were simply de-coupling caps and since the original design bypassed the two gain stages right between these to resistor cap combinations, if I didn't, the second cap becomes redundant. As for the size, I agree to be true to the original value it should be 0.1 uf.

As for the input resistor the original shows a 2.2M there. I am using a Tl072 IC which is a JFET input IC.

[Mark Hammer]

Here's an idea.  I shrunk the BYOC drawing down a bit and inserted a few new parts.  R7 has been dropped to 150k.  That gives the stage a max gain of 15x, instead of 22x, making a total circuit gain of 22x15=330, rather than 490.  So, still hot, but not quite as much.  R7 is augmented by part of a 100k pot.  When the wiper is all the way at one end, that 100k is added to the 150k, making the feedback resistance 250k, but just as importantly no resistance in series with the diodes.  So, set the gain high and you get hard clipping.  As you rotate the new pot in the other direction, though, you subtract feedback resistance, and add series resistance with the diodes to restore the softer clipping.  The new R6 sets a limit to how much series resistance.  I picked 8k2, because if you have the new pot set so that there is 70k on one side of its wiper, and 30k on the diode side, 70k+150k gets you 220k, and 30k in parallel with 8k2 gets you 6k5...close enough to the original specs.  Rotated such that the full100k is in parallel with 8k2, that gets us 7k6 in series with the diodes.  That, in tandem with less gain in that stage will get us softer distortion.

A simple, but interesting twist on a classic.  Might be worth a try.

So, after getting sidetracked by some MFOS modules, family events, the NHL playoffs, and home renovations, I threw together a little one-chip circuit last night that implemented what I describe here.  I'll try and finish the drawing and post later tonight.  I'm pleased to say the concept works, though I'm still debating whether it is better to have a pot that adjusts the series resistance with the diodes and overall feedback resistance in reciprocal manner, or simply a separate pot that adjust only the resistance in series with the diodes.  Achieving a suitable taper and dialability of "clipping-hardness" may be the chief determinant.

That said, varying the series resistance does produce audible and musically useful variations in hardness of clipping that can range from the familiar overdrive tone to very modest coloration.  Within the context of a Bluesbreaker, this would suggest that a 10k-20k pot, wired up as variable resistance, in place of the 6K8 resistor in series with the diodes, would make a very useful addition to the unit and expand its tonal palette.

[Gus]

When you have 4 diodes like used in a Blues Breaker you can do something that looks different
http://www.diystompboxes.com/smfforum/index.php?topic=103110.0
R13 is still in series with the diodes.  Also say if you want 6 diodes and symmetrical clipping you only need 5 the added diode goes at R13 position and you can still keep R13 in series with the new diode