Tweedle Dee PI - LNFB and trimmer

[Stephen1966]

Dumble's cathodyne, only real difference despite its apparent complexity, is in where it taps its output voltage of the cathode side of the circuit from. In principle, it is still the Ra, ra and Rk in series according to our formula. How we treat the series and Rk in particular though forms a special case.

Before proceeding any further, let's consider the equivalent circuits a, b and c:

Tweedle Dee - equivalent circuits.jpg

Rb in (a) sets the bias thanks to a tap for its accompanying grid leak resistor at its base and provides a little isolation from its accompanying power tube even when the wiper of the trimmer is set to 0% rotation, right up against foot of the bias resistor.

Rk2 (33k) was likely intended as an adjustment resistor during tune-up so in (b) Rt1 and Rt2 have been summed to create the fixed, final value, Rt = 89k. In (b) also, the trimmer has been divided into two co-dependant resistor blocks, Rtrim1 and Rtrim2. The trimmer is a linear 10k voltage divider so can be easily expressed as:

Vl = Vs (Vtrim2 / Vtrim1 + Vtrim2) ; where Vl = output voltage; Vs = source voltage.

We are interested in simply how it divides the resistance across it's rotation though so at 0% rotation it presents 10k which is added to Rt and when Rt = 89k, the sum is 99k. On the other hand, if the trimmer is set to 100% rotation, 0k are added to Rt and Rt remains at 89k. This is all fairly obvious and quite trivial given that the trimmer is linear and 10k.

It's easy to see that if the trimmer is set to 0% rotation in (c) then Rkb (the combination of Rb and that portion of the trimmer's resistance in series) is just 4.7k whereas if the trimmer is set to 100% rotation, Rkb has a value 14.7k and Rkt on the other side of the wiper has a value of 89k. Taken together and compared side by side, that represents a significant difference of voltage division. The grid bias is being tapped from the bottom of the (fixed) 4.7k resistor so should remain unchanged at its idle point and can be left out of the calculations at this point but what does matter is that there can be anything between 4.7k and 14.7k above the Rkt portion of the calculation and this variable should now be integrated in series with Ra and ra calculations.

I'm paraphrasing but Preisman's first equation (Eq.1) is the impedance looking into the plate terminal, Ap - excluding Ra. and Eq.2 is the impedance looking into the cathode terminal - excluding Rk - or Rkt as we have it here.

[Stephen1966]

It was already mentioned that the cathodyne naturally operates with 50% negative feedback so it follows that if we need to treat the Rk portion of the circuit as Rkt; that is, Rk = Rkb + Rkt, Rk - Rkb = Rkt then it follows that when we look at the impedance from Ap, we must also disregard Rkb from Rk but we can transfer this over to the plate resistor and effect a reduction in Rk from Ra. In simpler words, if we take Rkb from Rk we also need to take Rkb from Ra. Notice, the terms of the formula stay regular and the denominator doesn't change as it represents the total resistance in series, through the circuit. What we end up with is something like this:

Trimmer calculations.jpg

This is based on an ideal tube with ideal properties and a real tube is always going to be less than ideal.

Even so, this next slide from the FFT measurements indicate that the calculations are something like "in the ballpark", that both the plate and cathode outputs get smaller as the trimmer goes from 0% rotation to 100%, and that the voltages move in a parallel (more or less) direction. In these traces the green 'Pk-Pk [4]' represents the cathode and the red 'Pk-Pk [2]' represents the anode. The trimmer doesn't allow an equal output voltage for plate and cathode at any point in its travel.

Trimmer rotation.jpg

With the FFT measurements we have a glimpse of the next stage in the analysis of this circuit but the important thing to remember so far, is that in the present workings of the circuit values and their equivalent circuits we have been dealing with impedance. At this stage, before the reactance of capacitors starts to be taken into account, the resistances in the circuit can be seen as impedances and would yield similar results in either DC or AC. We aren't concerned with DC with regard to the trimmer though. It can be seen as a coincidence that some DC disparities can yield certain tonal profiles but it is not really the DC load that matters. In my hypothesis at least.

[Stephen1966]

As previously mentioned then, it's impossible to get an equal voltage amplitude when the cathodyne is unbalanced with a 110k resistor on the plate. It is possible however, if you replace Ra with a 100k resistor. This changes the harmonic distribution of the signal however.

Harmonics over 1kHz fundamental(2).jpg

There's more to explore with these charts such as how "linear" the non-linear distortion (harmonics) looks. The bigger effect on tone comes from the value of Ra with the more balanced outputs (Ra = 100k) creating a more even distribution of the even order harmonics. The second and third harmonics with the original Ra = 110k installed, are more tightly bunched together around the -40dB level whereas 4th, 5th, 6th and etc are under around -52dB. The sound floor sits around -70dB in, so well out of audible range. You can also see that the fifth harmonic swaps places with the third between the two versions - is there any tonal advantage in having the fifth harmonic more dominant than the third? You decide. If we were to plot a line at about 35% rotation vertically, we have a rough picture of where Dumble set his trimmer. Generally speaking, in addition to the spread of harmonic peaks you will notice the harmonics in Dumble's version are slightly more pronounced. The second harmonic sits around -38dB whereas the same is closer to -43dB when Ra has a more balanced 100k value. If we were to step back from the scrutiny of data points for a moment though, and look at this through half-closed eyes, you might see that when we have a 100k plate resistor the harmonic content is distributed across the bandwidth in a relatively linear and more or less even dispersion, but when the plate resistor of 110k destabilizes the output signals, the harmonic content is more clearly defined and peak-like in the same bandwidth.

[Edit: another slightly better way of thinking about it is that with the 100k plate resistor, the harmonics tend to ramp down in a relatively linear, diminishing fashion. In the 110k scheme though the second and third harmonics dominate and the higher harmonics step down more abruptly. Frankly, my ears are not the greatest but while the 100k version sounds slightly sterile, the 110k version sounds a little purer in tone. Something to think about...]

How I went about setting the trimmer deserves a special mention as well. The phase of the non-inverting and inverting signals should be as close to 180° as possible. The thinking is this. Pure sinusoidal waves are linear but carry a reactive component in the form of capacitance and inductance. The current waveform when the circuit is purely resistive, matches the voltage waveform with 0° phase angle between them, capacitive waveforms lag behind the voltage waveforms and inductive waveforms lead them. The phase angles between a current and a resistive waveform are mostly linear. All kinds of perturbations can introduce non-linear distortion into the signal though, harmonics are one form of non-linear distortion, intermodulation distortion is another, but they are all just distortion whether they sound pleasing or not. So far, I have been careful to keep the analysis of the cathodyne in the resistive domain. However, when you consider the reactance of a capacitor at different levels of current and voltage... and frequency, then it quickly becomes apparent that an educated guess becomes as good as any amount of number crunching. Capacitors are interesting like that and if you run these tests and tuning at anything other than 1kHz you will get different results each and every time. Just try dialling the signal generator down to 40Hz like in the Ampeg V9 procedure - nasty!

So why polar opposite outputs and why 1kHz? The frequency is just a handy benchmark that happens to sit well, within the spectrum of human hearing where we can distinguish spatial relationships. It's low enough to breed some quantifiable reactance in the capacitors and slow enough in the time domain (charging/discharging/riding the ripple) to generate some distinct non-linear distortion. Different frequencies, 800Hz, 400Hz have their proponents too, but I don't think it is too important. It would only be in the highest frequencies that the capacitors would act more like shorts and those are outside our area of interest, not to mention our hearing. There will be inductive loads in the circuit, and something as obvious a poor lead dress will have a significant effect on the sound floor and levels of distortion, usually not in a good way. The AC signal generator will have a low THD as well but this isn't hi-fi and a regular generator is going to provide as good as we need for this kind of work. What the 180° ensures is a benchmark mean. The cathodyne is meant to run with 50% negative feedback (we've been turning a blind eye to the LNFB circuit in the recent calculations, but only for simplicity's sake, it's still there!). In an ideal circuit with zero losses, the one side pulls at exactly the same time as the other one pushes. There is no lag or lead and so when one peaks, the other does too. By removing one of the variables (voltage phase) it is easier to see and evaluate the effects of the other variables.

The oscilloscope probes go one each after the coupling caps to the power tubes, but before their grid leak resistors and another goes on the resistive speaker load. The probe on the speaker load is the source for the FFT and the probes on the PI's cathode and anode outputs are the source for the phase measurement. The latter will also give us voltage measurements as well but these are not that important. Using a 1kHz, 100mV(rms) signal into the front of the amp, the channel volume is raised until the non-inverting and inverting sides of the cathodyne are 180° in phase. With a 100k plate resistor, the volume level is around 4 on the dial that goes to 12. With the 110k resistor, the volume level is around 3. After measuring and tuning by eye, we must finish off with tuning by ear.

[Stephen1966]

I think once you realise the goal is not to get an equal voltage amplitude from the plate and cathode, but to trim it up to get the most differential power from the fundamental it liberates you to think about the tone of the amp in a new or at least different way. That word "differential" has gotten me into trouble before now, but here, I mean it as the measure of the effective power between the outputs. The last chart showed the effect of biasing the cathode vertically so to speak, for more or less tight aggregates or spread of harmonics. Sticking with the deliberate imbalance of the 110k plate resistor and biasing the cathode around the narrow furrows carved out of the bandwidth between 25% - 50% rotation we can now see how that pans out in a horizontal manner as we go from 1 on the channel volume all the way up to 12. This time extending the analysis up to H8 and correlating the volume settings with phase angle readings.

Harmonics data FFT 09.jpg

This is as graphic a representation of the tone of the amp as you are likely to get - if anyone wants to crack open a real TD... please! But it all relates to what we typically see in this kind of amp. The amp really doesn't start to kick in until around 2 on the dial. Around 3 are some of the sweetest tones great for recording and with tons of headroom. A typical 5E3 gets gnarly in pretty short order but Dumble's mods extend the headroom into the higher gain region and with a test signal of 1kHz at just 100mVpk-pk as here, we can drive that all the way up to clipping which sees its onset just before 10 on the dial. The tone gets much more compressed from there to the end of the scale and with input signals greater than 100mVpk-pk, there is typically a sag between 11 and 12 which also, typically, cuts the high end as well. It's around this level as well that the 5E3 is notorious for blocking distortion. Mine does that as well if I drive it too hard but my top end doesn't drop out, more like it goes into hyper-saturation mode as the effect of the LNFB cuts out and the cathodyne pumps out every ounce of gain it has held in reserve. That's abrupt, the signal clips hard and if you stick with MrD's grid bias point set by the 4.7k bias resistor, the grid voltage will clamp as well. Not unusable but you have to be in a special kind of mood to linger any amount of time there. Dialling back the guitar volume in these higher gain situations cleans the signal up significantly in inverse proportion to how hot your pickups are.

Tone and power-wise, around 9, just before the onset of clipping shown here, is the ideal with a grunt and a growl that will scare your neighbours and any passing small children. Saturation is full and open-throated with strong low-to-mid mids (!) and highs are articulate and sharply defined. In the "old days" I didn't have any gear to adequately record anything with volume beyond 2, but now I have no excuse, I am just being lazy. I will get around to it before all this is done.

When it comes to biasing the trimmer with signals at 180° you may find as I did that you only need to advance the trimmer a little more towards the tail to really bring out the beats on the harmonics using the standard repertoire of listening tests discussed at length elsewhere on this forum. That swirling, touch-sensitivity and the slightly out-of-phase bloom it brings. The signals are very stable around 3 and 4 on the volume dial and then gradually drift more and more out of phase until around 10 when clipping, the hyper-saturation, and/or sag, start getting wild and the phase swings on a knife edge between symmetry and asymmetry. My understanding of this is that as the power of the harmonics increases so do the non-linear power resonances they create. Like dropping a stone into still water and watching the ripples radiate outwards, then dropping another stone and watching how the ripples merge and coalesce around nodes of convergence. It's possible to get the phase symmetry such that this does not immediately result in intermodulation distortion but instead offers a cleaner focused and pleasant sounding distortion across a wide bandwidth.

[Stephen1966]

I found an interesting paper that offers a more accurate mathematical model of the cathode output when it is taken from bottom of the bias resistor. Rightly or not, many of the small signal models for the cathodyne out there, are predicated on the assumption that we want a more or less perfect balance between the outputs and in general, the value of the bias resistor is either ignored (as a significantly smaller value than the tail or plate resistors) or, in more hi-fi settings, summed with the tail to provide an equivalent load alongside the plate resistor. Check out the example of the cathodyne @TenOver graciously provided. In my mind, it's not enough to brush the bias resistor aside. In more conventional settings the bias resistor is very small proportionally with the tail, 1.5k:56k in the 5E3 but in the TD as part of the equivalent circuit in which the trimmer is subsumed in the bias and tail resistors it can clock in at 14.7k:89k. These numbers are much more significant than the Fender's and to ignore them is a fudge too far. So Aaron's paper is a valuable addition to the field but it only goes as far as the general models of vacuum tube amplifiers, the TD (by design) represents a special case, and deserves a treatment that acknowledges it as such.

This paper, by Aaron Lanterman is a little more developed than his YouTube videos (which I urge you check out as well) and using the Thevenin model for small signal analysis it does a great job of covering the cathode follower with details of mathematical models for voltage and impedance. I cannot say why he didn't actually cover the models for the cathodyne but in his treatment of the cathode follower there is an analogous treatment almost ready to go, and in section 3.4 where he discuss the long-tailed pair he suggests following the models for the CF for the cathodyne. I was able to do that but I ran into some problems when I tried applying his method to the voltage equations for the plate outlet.


https://www.mdpi.com/2079-9292/12/23/4804

electronics-12-04804.pdf

These are the equations I came up with.

Voltage and Impedance formulas.jpg

Eq.1 is really no more than an algebraic reordering of the equation I offered earlier. This had to different from Lanterman's because all the experimental test data showed that as the value of the voltage divider on the cathode led to a decrease of the cathode voltage output, the higher plate voltage output followed suit and also decreased accordingly in a more or less fixed proportion with the cathode. Lanterman suggests that by adding "R_L" to "rp" (Ra + ra in our designation) to his CF formulas (Eq.10 and Eq.11) they can accommodate the cathodyne model but if we do this with his first equation (Eq.10) the results remain constant for all the variables of the bias and tail resistor and the result is lower than for the cathode at the same trimmer setting. The problem with Eq.10 is that the sum of the bias and tail resistor appears in both the numerator and the denominator - admitting no variation - whereas the addition of the plate value to the denominator makes the denominator bigger (obviously!) and that makes the fraction smaller compared to its counterpart for the cathode! Eq.11 has a similar problem in that again, the addition of plate resistor value in the denominator results in a smaller fraction compared alongside its counterpart.

Lanterman equations.jpg

However, when we substitute the tail value with the plate value in Eq.11, we get the desired higher value result but again, it's a constant so I added the variable bias resistor which is a negative value because it is on the other side of (as he terms it) of the voltage-controlled voltage source (VCVS). I cannot say for certain that this is the correct model to use or if the formula is absolutely correct but if it's a fudge, it's pretty close because after literally hundreds of measurements were made the formula is working.

Voltage calculations.jpg

I am a little suspicious of carrying Lanterman's directive through and applying the same methodology to the output impedance calculations.

Impedance calculations.jpg

The results here are much higher because of the bias resistor in parallel with the tail. Instead, I would be inclined to go with Blencowe's formula for the effective 'differential output impedance' and devise a workaround for the problem of the old predicate, Ra = Rk = R.

PI equations (1) - differential output impedance.jpg

Even considering the results on the high side though the output impedance is still somewhat lower with less effect on higher frequencies in series with the coupling cap than would be seen in an ordinary common-cathode amplifier.

[Stephen1966]

For the sake of completeness, here is the RDH4 entry for the cathodyne:

RDH4 - cathodyne phase splitter.jpg

You will notice Eq.34a is the source of Blencowe's equation for the effective output impedance. I am inclined to go with this rather than Lanterman's suggested equations. It also doesn't really matter if you use R_L (Ra = 110k) in this formula or the derivative R when R = Ra + Rk / 2 = 106.85k both numbers provide Zo (effective) = 900 Ohms. Though, oddly enough, if you subtract the the Zo (cathode) from the Zo (anode) in Lanterman's equations, the differential is around 900 Ohms in each case. In any case 900 Ohms is very good for our next stage and the Zo is so negligible, Blencowe suggests we might ignore it when calculating the value of the coupling capacitor to the power stage. We already know the value so we can use this with the Amp Books, Coupling Capacitor Calculator https://www.ampbooks.com/mobile/amplifi ... capacitor/

Plugging in our component values, we have this:

FC of cathodyne output stage.jpg

The output impedance of the cathodyne and the input resistance of the following stage are essentially a voltage divider and the gain is almost unity gain representing such a small loss at 82 Hz that the Zo might, as Blencowe suggests, be ignored in all practical use scenarios.

I have led us down this garden path of small-signal analysis because I wanted us to understand the part impedance plays in the operation of the cathodyne and in the setting of the trimmer. The impedance has both a real part, resistance, and an imaginary part, reactance. With resistor networks, the resistance and the impedance of the circuit are barely any different and as far as phase angle relationships are concerned both sit at 0° relative to the AC voltage. I am going to make a mess of it if I try to explain the mathematics of it but at the bottom of the page in this link there is a simulator you can play with.

https://www.intmath.com/complex-numbers ... -angle.php

Things, as I mentioned earlier, get interesting when you start factoring in the reactance of capacitors and to this we might also add the internal capacitance(s) of the triode as well as those of the couplers in the following stage. Because of the effect of AC frequency on a capacitors reactance and of its reactance on the impedance and of the impedance on the phase angle, it would be somewhat futile to pursue any further, the calculations of complex numbers and phasors in any practical, meaningful sense. To understand that they exist and that they have a complex interaction with other non-linear expressions of power (harmonics) is enough to base our understanding of the function of the trimmer in a more or less quantifiable context. It also tends to qualify the imbalanced cathodyne as a generator of unequal voltages and the behavior of the harmonics across a range of power settings.

I've heard it said that the trimmer serves no real useful purpose and granted, it has only a very minimal affect on the harmonic profile, changing the values of the plate or tail resistors has more impact, but if you know what you are looking and listening for, finding it becomes so much easier. An oscilloscope with FFT is a very handy tool for locating the sweet spot when biasing the trimmer but is it indispensible? No. Not for our purposes. Yet again, the organs on the sides of our head can tell us all we need to know, I think it takes some strong critical listening skills though. If you are going to pass the tone off as "good enough" then you might never hear the difference or know what you are missing. But as I said, if you know what you are looking for...

Finally, don't say I didn't warn you...

Sound test.mp3

[Stephen1966]

I recently had to reset my PI trimmer and what I found was instructive.

The context here is that I want to try something like a cold-clipper/lead channel mod on the normal channel. More actually an asymmetric bias on the cold side. The cold-clipper is quite extreme and in the parallel first stage is asking a bit much of this economical circuit and the lead channel while introducing a little cooler bias seems to more aimed at increasing the gain, which I didn't want. So I played around with a few different bias options and settled on Rk = 2k7 and Ck = 4.7uF.

My first stage B+ is almost exactly the same as the original's at 286.5V

V1 Tweedle Dee - 12AX7 chart.jpg

V1b 5E3-D (BRIGHT) - 12AX7 chart.jpg

The performance of this channel is almost the same as the original on paper, surmised from the data we have. They are roughly centre biased.

The change in bias on the normal channel shifts the operating point right but while the voltage gain (blue lines) falls, the maximum voltage swing (orange lines) increases. So an even cleaner channel.

V1a 5E3-D (NORMAL) - 12AX7 chart.jpg

The change in tone is most apparent though. This is a much smoother, full-bodied, bluesy kind of sound. Tubey? Yes but requires the neck pickup. It's still very touch sensitive but that is improved with the trimmer setting as well. It will overdrive at a fair clip when the volume is up to 7ish but lighten the touch and cleans up superbly.

[Stephen1966]

Here are the different trimmer settings from 0% to 100% rotation. Again, the volume on the channels was kept fairly low so that the opposing signals off the cathodyne were 180° apart (between 3 and 4 on the dial that goes up to 12) but with the FFT signal being measured at the speaker output. Unless you have a really expensive scope, you are only going to analyse the FFT off one channel of the scope.

This first graph shows what I think a typical Tweedle Dee harmonic profile would look like. This is an earlier snapshot which I included earlier and I started modifying my amp's circuit shortly after. The cathode bias resistor for the power tubes was changed from 250 Ohm to 270 Ohm (increasing the B+ voltages all the way across the dropping string) and I reverted from the GZ34 back to the 5Y3 (which dropped all the voltages across the dropping string) in order to tame the voltages. Together, these two changes brought the voltages almost perfectly to the same voltage as measured by Charlie at the first stage of the preamp. This graph, was made when the voltages were about 15V higher. The 1kHz signal was at 100mV(rms) which is quite high, and sent through Normal, channel 1. The other knobs are held at 50% rotation as per the Fender test conditions.

Harmonics with Rk 1k.png

After my mods I injected the same signal into the Bright channel which retains the Rk = 1k and 25uF cap as per the TD, with the PI outputs again set to 180° and this produced these results:

Bright - Harmonics with Rk 1k.png

I was really curious to see what the harmonic profile of the normal channel would look like now with Rk = 2k7 and with a 4.7uF cap.

Normal - Harmonics with Rk 2k7.png

I use the simple test of playing notes d, d# and e on the second string and I listen for the beats. When the harmonics are very closely bunched and when they appear to cross-over as they do in the first graph at about 35% rotation mark, that's when the beats can be heard. Some people call it a swirling as well, and there is kind of bloom of the upper partials where the fundamental rings first and then the higher octave harmonics start to ring out, sometimes, with the d# note for instance they are hard to hear. Acoustically, the instrument does not freely resonate at d#. However, a dissonant (odd order) harmonic can eventually be heard if you listen carefully for it.

Is it a coincidence that Dumble's trimmer was set like this?

Untitled-1.jpg

This crossover point or just slightly in advance of it, seems to consistently yield the "correct" bias point. My second chart showing for the first time, the signal injected into the bright channel reveals more or less equal levels for all harmonics after the third - as can be seen in this screenshot from 50% rotation:

SDS00649.jpg

The third harmonic dominates but all the other higher peaks are within around 10 dBV of each other. The equal level nature of the peaks has a very different character to the Normal channel at the same trimmer setting:

SDS00614.jpg

The gradually decaying peaks present a much warmer, fuller sounding tone. It isn't all about the even ordered harmonics. The even ordered harmonics, H4, H6, H8 and H10 are all at close enough to 40 dBV here the odd ordered harmonics dominate and this is not an unpleasant tone at all. It's reminds me of BB King.

It was very interesting to discover that when I was biasing the normal channel the sweet-spot where the harmonic beating becomes audible is around 60% rotation, right where the higher order harmonics cross over. I think it only takes a very small difference in phase angle in this area to get the harmonics oscillating/interacting with one another. The beat and the higher harmonics emerge quite readily - it's deep inside the structure of the notes - not quite the same as when we hear the beats between two slightly out of tune fundamentals when we are tuning up. The fundamental remains clean, its the overtones which perform this merry dance.

[Stephen1966]

Clipping versus saturation - includes sound samples modelled, so you don't have to!