Charles Eric LaForest, PhD., GateForge Consulting, Ltd.
These are quick notes and experiments to flesh out an idea I stumbled upon. It's a detailed draft, not a full treatment, of a more detailed design approach with diodes for distortion and saturation.
You can try out these circuits (including the bias supply and input buffer) in the interactive CircuitJS1 simulator: germanium_replacement.cjs1.
Germanium diodes create a pleasant, softer distortion than modern silicon diodes, but are no longer in production, are getting hard to obtain, and there have been cases of Schottky diodes being passed off as germanium diodes, both deliberately and accidentally.
Both Schottky and germanium diodes have similar lower forward voltages at currents in the range of milliamperes, but Schottky diodes are much leakier when reverse-biased, and so do not behave the same when used to clip signals.
Also, approximating diodes as one-way switches with a relatively constant forward voltage drop misses out on their best feature for creating pleasing distortion: the non-linear (exponential) relationship of the forward current to the forward voltage.
Both germanium and silicon diodes are modeled by the same Shockley diode equation, so we should be able to find a different operating point for a given circuit so that a silicon diode will give us the same behaviour as a germanium diode.
I'm not going to try and analytically derive the desired behaviour from the system formed from the Shockley equation along with the voltage divider and parallel resistance equations. Instead, it's much easier to experiment directly in an interactive simulator and explain qualitatively what happens and why. That will suffice for you to alter these examples to get the specific behaviour you want.
Let's assume a common signal IN, already pre-amplified and biased, which we use to drive different circuits. The first one is the reference: a plain non-inverting amplifier with a gain of 2. The op amp output feeds a voltage divider formed by the two 330k resistors, with the "top" resistor driven by the op amp, and the "bottom" resistor going to the virtual ground (Vbias). We use the de-biased output (OUT_REF) as the scale reference so all other circuit outputs are comparable.
We will add diodes to the reference circuit, from the op amp output to its inverting input, to construct a conventional TubeScreamer-style soft-clipping circuit, where the "top" and "bottom resistors each control a different aspect of the resulting distortion.
The "top" resistor is in parallel with the dynamic resistance of whichever diode is forward-biased at a given moment. The ratio of these parallel resistances determines what fraction of the total current goes through the diode, and thus how much the non-linearity of the diode affects the total current.
The "bottom" resistor is in series with the above parallel resistance, and thus both limits the current through the diode and converts that current into the feedback voltage controlling the op amp. A larger resistance multiplies more the changes in current through the diode, but also lowers the current. By choosing the right resistance value, we can tune the non-linearity of the diode.
We will make two soft-clipping variations of the reference circuit: one with 1N34A germanium diodes (OUT_IN1N34A), and one with 1N4148 silicon diodes (OUT_1N4148). These two circuits will of course have different clipping behaviour.
To make the silicon diode circuit clip identically to the germanium diode , we take the same silicon soft-clipping circuit, but reduce both "top" and "bottom" resistance by 100x from 330k to 3.3k (OUT_1N4148_SOFT). This dual reduction leaves the maximum gain unchanged (2) and accounts for the lower dynamic resistance of the silicon diode.
(Addendum: later experiments showed a 70.2x reduction (4.7k resistors) gives an even closer match.)
The waveform on the left compares the four circuits. The largest, green trace is the undistorted reference output (OUT_REF), which reaches 1.53Vp, implying the input signal (IN) reaches 765mVp.
The smallest, red trace is the silicon soft-clipping output (OUT_1N4148), which has the most clipping since the 1N4148 diode has the least dynamic resistance, thus lowering the overall gain as part of the "top" parallel voltage divider resistance. We know this curve is clipping and not just a lower gain, because the slope near the zero crossing (where the diodes conduct very, very little) is the same as the reference.
The two middle, nearly overlapping purple and orange traces are the germanium (OUT_1N34A) and germanium-replacement silicon (OUT_1N4148_SOFT) soft-clipping circuits. Although I show only one operating point, the two circuits remain matched regardless of the input voltage (IN) or the circuit gain. Any proportionate change (e.g.: halving, tripling, etc...) to the "top" and/or "bottom" resistors of both the germanium and germanium-replacement silicon circuits results in the same matching waveforms.
Given a constant amplitude input, changing the "top" or "bottom" resistances has different effects on the overall distortion:
Increasing the "bottom" resistance decreases the current through the forward-biased diode, which lowers its forward voltage, and vice-versa when decreasing the "bottom" resistance. This change affects the signal level at which clipping begins, where the gain smoothly transitions from the clean undistorted gain set by the "top" and "bottom" resistors, to the non-linear, distorting, reduced gain from the diode dynamic resistance in parallel with the "top" resistor.
Decreasing the "top" resistance lowers both the clean and distorting gains and thus also lowers the peak voltage, which means less of the waveform is distorted by clipping, as set by the "bottom" resistor.
Increasing the "top" resistance increases mainly the clean gain, giving a sharper slope to the unclipped signal below the level set by the "bottom" resistor, but without changing the peak voltage significantly since the "bottom" resistor sets the diode current.
Varying both "top" and "bottom" resistors together, while keeping the gain constant, changes the amount of saturation (decrease in gain past a certain amplitude, as in tube amplification or tape recording or transformer saturation, etc....) without adding higher overtones from conventional gain-based distortion as when increasing the "top" resistance only.
By selecting a mild gain and the right input level and "bottom" resistance, this saturation behaves like a mild compressor (e.g.: 2:1 or so) with an immediate attack and release, which is perfect for adding pleasant overtones to the lower notes of a bass guitar while controlling their usually higher amplitude. It also limits sharp attacks from tapping, slapping, and picking.