Condor Cab Sim/Bridged-T filter calculations

[jrc4558]

Good day to you all!
First of all - I finally finished Condor ( http://www.runoffgroove.com/condor.html ) and am altoghether pleased with the sound. Hopefully I will be able to post some sound samples soon.
I tried implementing the 'marshall curve' mod suggested at ROG website. It does what I want it to, but does too much. Mids are overpowering the whole spectrum and the sound is muddy. I really like the 4*10 setting though. So, I decided to investigate the bridged-t circuits.
It's either late, and my cognitive functions are depleted after finishing a research paper proposal, or I'm just a poor hunter, but...
...can somebody please tell me what the formula for the bridged-t filter is?
I promise I'll get a book from the library next week and read on it, but for now...

[Rob Strand]

Use a circuit simulator to play around with the parts.

The equations for the bridge-T network can be written down but they are complicated and not revealing to most people.   The circuit in the Condor has the added issue that it's not a straight forward Bridge-T.    The 100k to Vref complicated things and so does the 20k source impedance from the previous JFET stage has an effect.  Incorporating these two aspects into equations results in big mess.  Playing around on the circuit simulator is much easier.

[MartyMart]

Without doing the "Maths" I followed the ROG site advice, plus a tip from stm and did the
following tweaks :
Two 47n's between U1A/U1B down to 33n
Four 22k's from filters at U2A/U2B down to 15k and the coresponding 3n9 caps
to 3n3.
I also lowered the first 47n off Q1 to 33n
Result is more like a 2 x 10 I guess, but I liked the increased brightness and reduced
low end a lot.
MM.

[jrc4558]

Use a circuit simulator to play around with the parts.

Thanks for the advice, but I never used a circuit simulator! What are they? Which one do you use?

[stm]

...can somebody please tell me what the formula for the bridged-t filter is?
I promise I'll get a book from the library next week and read on it, but for now...

I developed the design equations two years ago from now.  Check here:
http://www.diystompboxes.com/smfforum/index.php?topic=25788.0
Not to be found on a book, AFAIK.  

[jrc4558]

Without doing the "Maths" I followed the ROG site advice, plus a tip from stm and did the
following tweaks :
Two 47n's between U1A/U1B down to 33n
Four 22k's from filters at U2A/U2B down to 15k and the coresponding 3n9 caps
to 3n3.
I also lowered the first 47n off Q1 to 33n
Result is more like a 2 x 10 I guess, but I liked the increased brightness and reduced
low end a lot.
MM.

That's the thing! I want to switch between modes and want to have to use a 3pdt maximum.

[jrc4558]

I developed the design equations two years ago from now.  Check here:
http://www.diystompboxes.com/smfforum/index.php?topic=25788.0
Not to be found on a book, AFAIK. 

Yes, I saw those! I don't quite understand the notation of it. What's A with an "o-like" symbol above it? 

[stm]

I developed the design equations two years ago from now.  Check here:
http://www.diystompboxes.com/smfforum/index.php?topic=25788.0
Not to be found on a book, AFAIK. 

Yes, I saw those! I don't quite understand the notation of it. What's A with an "o-like" symbol above it? 

Those A's with the "o" on top are for Angstroms.  FYI, one Angstrom equals to 1 nanometer (a really small distance).

Why in hell are those symbols in there?  I just don't know.  They weren't when I did the original post.  Maybe they appeared during one of the server changes due to a text conversion issue.  Nevertheless here is an updated version of the text:

-------------------------
0) Definitions & Circuit
-------------------------

   fn    : notch frequency in Hz
   depth : notch depth or attenuation in dB

   Bridged Tee circuit:
                          C1
                          ||
                  .-------||--------.
                  |       ||        |
                  |                 |
                  |   ___     ___   |
             In o-+--|___|-+-|___|--+-o Out
                       R1  |   R2
                           |
                          ---
                          --- C2
                           |
                           |
                          ===
                          GND

   (created by Andy´s ASCII-Circuit v1.27 beta www.tech-chat.de)

----------------
1) General Case
----------------
                  1
   fn = ----------------------   [Hz]
        2·PI·sqrt(C1·C2·R1·R2)
       
                 (    C1·(R1+R2)      )
   depth = 20·Log(--------------------)  [dB]
                 ( C1·(R1+R2) + C2·R1 )
             
----------------------------
2) Simplified Case: R1=R2=R
----------------------------
                1
   fn = ------------------   [Hz]
        2·PI·R·sqrt(C1·C2)
   
                 (    2·C1   )
   depth = 20·Log(-----------)  [dB]
                 ( 2·C1 + C2 )

---------------------------------
3) Step-by-Step Design Procedure
---------------------------------

   a. Choose C1 to your liking (1nF to 10nF are good starting values)
   
   b. Choose C2 for the desired notch depth according to the following table:

         C2     depth [dB]
      ---------------------
        1 x C1      3.5
      1.5 x C1      4.9
      2.2 x C1      6.4
      3.3 x C1      8.5
      4.7 x C1     10.5
      6.8 x C1     12.9
       10 x C1     15.6
       15 x C1     18.6
       22 x C1     21.6
       33 x C1     24.9
       47 x C1     27.8
       68 x C1     30.9
      100 x C1     34.2
   
   c. Calculate both resistors as:
 
                        1
      R1 = R2 = ------------------   [ohms]
                2·PI·fn·sqrt(C1·C2)

      where fn is the desired notch frequency in Hz

-----------------
4) Mods & Tweaks
-----------------

   a. Frequency tuning

      If you move both resistors equally you change notch frequency
      without altering the notch depth. A dual ganged pot can be used
      for this purpose. Don't forget to add a series resistor on each
      pot to limit minimum value.

      My recommendation here would be: a dual ganged 100k LOG taper +
      1kohm in series with each pot, allowing a frequency tuning range
      of 100:1.

   b. Notch depth adjustment

      You can change the notch depth WITHOUT altering the notch frequency
      if you change the C1/C2 ratio without altering their product (i.e. C1·C2).

      In plain english, choose a constant K and then divide C1 by K and
      multiply C2 by K.  This effectively changes the notch depth keeping
      its frequency intact.  K values greater than one increase original
      notch depth, while values smaller than one reduce its depth.

      There is no easy way to accomplish the above with tunable elements,
      so the best you can do in this respect is have a two or three position
      switch to select from some preset depth values.  A six position rotary
      switch would be ideal.  By the way, to implement zero dB notch depth
      or all-pass response you just have to leave C2 open.

   c. Use of two independent pots for R1 and R2

      This offers an interesting option to control both frequency and depth
      but in an unusual manner.  Assuming identical valued pots and series
      resistors the situation is as follows:

      * Whenever both pots track each other (on the same value), you will
        be changing frequency maintaining the original notch depth.
             ___        ___
            / | \      / | \
           |  '  |    |  '  |
           | R 1 |    | R 2 |
            \___/      \___/

      * If R1 pot is greater than R2 pot you will increase notch depth with
        respect to the original desing value.
             ___        ___
            /  /\      /\  \
           |  '  |    |  '  |
           | R 1 |    | R 2 |
            \___/      \___/

      * If R1 pot is less than R2 you will reduce notch depth with respect
        to the original design value.
             ___        ___
            /\  \      /  /\
           |  '  |    |  '  |
           | R 1 |    | R 2 |
            \___/      \___/

   d. Low-frequencies with more attenuation than high-frequencies

      Place a parallel resistor with C2 (let's name it Rp).  This attenuates
      low frequencies, producing an overall increase in the high frequency
      content.  The resulting frequency response of this filter is akin to a
      typical Fender or Marshall tonestack in the sense that you have a
      notch and low frequencies are more attenuated than high frequencies.
      There are two advantages in this:

      * The circuit has minimum possible attenuation, since very high
        frequencies pass directly through C1 (0 dB).  On other tone stacks
        you can have an overall insertion loss in the range of 6 to 20 dB!

      * You only need two capacitors and three resistors to implement a
        fixed-setting typical tone stack response.  On the other hand, if you
        replace values on a standard tonestack, you may need up to three
        capacitors!

      The attenuation suffered by the low frequencies now is:

      AL = 20·Log[ Rp / (R1 + Rp) ]

      For instance, if Rp is nearly half R1, the lows will have approximately
      10 dB attenuation with respect to the highs.

   e. High-frequencies with more attenuation than low-frequencies

      Place a series resistor with C1 (let's name it Rs), thus attenuating
      high frequencies while letting low frequencies pass without attenuation.

      This could be useful at the output of an overdrive type circuit, where
      some high-frequency attenuation might be desirable apart from the
      mid scoop or notch.

      The attenuation suffered by the high frequencies with respect to the
      lows is:

      AH = 20·Log[ R2 / (R2 + Rs) ]

Note that when adding Rs or Rp the depth of the notch and its frequency are somewhat
affected. Nevertheless the proposed design method still provides a good starting point.

[jrc4558]

Thanks for the write-up! I already printed it out and it goes into my folder for sure!
One last question: resistance in Ohms, capacitance in Farads?

[stm]

One last question: resistance in Ohms, capacitance in Farads?

Yes, thus 10k should be written as 10000 and 22n should be written as 22E-09 in the calculator or spreadsheet.