Fuzz Face DC Bias

[mac]

Finally I have some time to make public the equations I used when I compiled a little app I wrote some years ago which calculates the Fuzz Face DC bias. The app is now updated to include the effect of germanium leakage current.
Suggestions and corrections are welcome.
In case text is not displayed good go to:

http://geocities.com/guitarfxs/docs/fuzzface_dc_bias.pdf

The Fuzz Face DC Bias Problem:

The Fuzz Face is by far the most simple and good sounding pedal I've ever heard. That's why I am trying to describe its DC behaviour in cold numbers which I guess some elect. eng. will understand better than me! I hope that it also could lead to a better AC understanding. 

The DC circuit is shown below. It is very simple. One transistor (Q1) DC coupled from its collector to the base of a second transistor (Q2) which provides some DC current from its emitter to the base of the first transistor. A first look suggest that the circuit is unstable if Germanium transistors are used. These transistors are very much sensitive to temparature changes than Silicons ones. Any drift caused by temparature in Q1 will be amplified by Q2 and will change Q2's bias drastically. This is the major drawback of the Fuzz Face.

The AC analysis and history can be read from RG Keen article found at www.geofex.com

To attempt a solution that takes the effect of leakage into account I will asume that the current flowing through the collector is ic* = hfe*ib + iL, where ic, ib and iL are the collector, base and leakage current, and hfe the internal gain of the transistor. Similar the emitter current is ie* = (hfe + 1)*ib + iL. Note that ib = ie* - ic*. I used the same assumption to describe RG Keen's Germaniun transistors test in my post at www.diystomboxes.com forum.
To simplify things, I also assumed a fixed value for the base-emmiter Vbe voltage drop.

With the above assumption the equations for this circuit are:

[1]  i1 = ic1* + ib2
[2]  ie2* = i3 + ib1
[3]  vcc = i1*R1 + veb2 + i3*R3
[4]  i3*R3 = ib1*R4 + vbe1

We want to calculate the currents and voltages as functions of R1, R2, R3, R4, hfe1, hfe2, vcc, iL1, iL2, vbe1 and vbe2.

Let's start by replacing i3 in eq [4] using eq [2]

[5]  (ie2* - ib1)*R3 = ib1*R4 + vbe1

Noting that ie2* = (hfe2 + 1)*ib2 + iL2 we have

           (hfe2 + 1)*ib2*R3 + iL2*R3 - vbe1
[6]  ib1 = ---------------------------------
                      (R3 + R4)

Using eqs [1] and [4] into eq [3], i1 and 13 can be eliminated

[7]  vcc = hfe1*ib1*R1 + iL1*R1 + ib2*R1 + vbe1 + vbe2 + ib1*R4
      
         = ib1*(hfe1*R1 + R4) + ib2*R1 + vbe1 + vbe2 + iL1*R1
        
ib1 in eq [7] can be replaced using eq [6] to finally have ib2 in terms of known values. After a little algebra

           vcc - vbe1 - vbe2 - iL1*R1 - (hfe1*R1 + R4)*(iL2*R3 - vbe1)/(R3 + R4)
[8]  ib2 = ---------------------------------------------------------------------
                             R1 + (hf1*R1 + R4)*(hfe2 + 1)*R3
                                  ---------------------------
                                          (R3 + R4)

Now that we have calculated ib2, the rest of the currents can be computed easily. First ib1, then i3 and i1. But we are more interested in the collector voltage of both transistors, vc1 and vc2

[9]  vc1 = vcc - R1*{hfe1*(hfe2 + 1)*R3 + 1}*ib2 - R1*{hfe1*(iL2*R3 -vbe1) + iL1}
                     ------------------                     ------------- 
                         (R3 + R4)                            (R3 + R4)
                       
[10] vc2 = vcc - hfe2*ib2*R2 - iL2*R2

It is not evident at first sight the behaviour of vc1 and vc2 when  hfe1, hfe2, iL1 and iL2 change with temparature. Most likely, and I am playing by ear here, hfe and iL change in the same proportion with temperature. An increase of temperature will move vc1 slighty down but vc2 can go up significantly.
If you could heat one transistor and keep the other at constant temperature you will notice an opposite effect on vc1 & vc2. If hfe1 increases then vc1 decreases and vc2 increases, while If hfe2 increases then vc1 increases and vc2 decreases. But the effect of hf1 is larger than hf2.
As you can see from eq [8] ib2 is very sensitve to small changes in hfe1 and iL1 as expected.

You can put this equations into a Excel or just write a little program to compute vc1 and vc2. Or simple go to:

http://geocities.com/guitarfxs/exes/biascalculator.zip

This small program simulates de dc behaviour of the Fuzz Face, and voltage divider and Big Muff-like stages.

mac