07 January 2016

cap decoupling of triodes /4

This is the fourth and final part of the lecture that I held at the 2015 European Triode Festival (ETF). It is about new insights, new graphs and new formulas that I developed during 2015 for cap decoupling of resistance-loaded triode gain stages.

note: if the terms ‘right-sized time constants’, ƒinterest, ƒ-3dB, hippies, or ‘Nixonian decoupling strategies’ do not sound familiar to you, then I recommend that you start off by (re)reading part one. If the term Cideal does not sound familiar, (re)read part two. And if the terms Zpin, Cpin and (Rk-compensated) Ck do not sound familiar, (re)read part three.

In this instalment we will look at triode gain stages with both a B+ and a kathode cap:

Did you just yawn? I do not blame you, because it does not get more standard than this. There are a gazillion schematics—single-ended, push-pull, off-the-wall—that use this gain block. Ironically, this ‘boring’ set up is more complex than the ‘interesting’ circuits we focussed on in part two and three.

when two is one

First things first: with two caps, does this circuit have a second-order roll-off? No, it is first-order, because the two caps simply work in series:

The lower (kathode) capacitor may be in parallel with a resistor and these two impedances get multiplied by µ+1 by the tube—but two caps in series it is.

This is the moment we are going to reap the profit of all the work we have done in part two and three of this series. Knowing that Cideal is the, ehm, ideal solution for the B+ cap, and Ck is it for the kathode cap (already incorporating µ and compensation for Rk), the act of putting them together is as simple as that of any two series caps:
C = A × Cideal, Cdk = B × Ck, where 1⁄A + 1⁄B = 1
Cdk stands for the definitive kathode capacitor value.

Normally, a numbers game would now ensue. Out of the thousands of (A, B) combinations possible, each of us would pick some aesthetically/mathematically pleasing one, e.g. (A, B) = (2, 2) or the one where C and Cdk end up with the same value. But then, we remember what we learned in part two: ‘In most practical circuits, the B+ capacitor will be 10 to 100 times larger than Cideal.’ Well, that settles it then. Instead of performing all kinds of stunts to get C in the direction of Cideal, I say we go with the flow, let C land where it does in practice and calculate the definitive kathode capacitor as follows:
Cdk = Ck / [1 - Cideal/C]
With C = {10–100} × Cideal, Cdk will range from 1.11, down to 1.01, times Ck, i.e. basically Ck. We observe that because in most practical circuits the value of the B+ capacitor is out of control, it is the kathode capacitor that is in control of the time constant of the stage. It is therefore very ironic that there was no widely publicised, correct formula for the kathode cap value. That is, up to now.

going down

Let’s have a look what is going on. The orange curve stands for the time constant you are trying to achieve:

(x: frequency in octaves; y: decibels)

Here are response curves for the practical-circuit range of B+ capacitors:

From right to left: C = {10, 10√10, 100} × Cideal.

And here are response curves for the practical-circuit range of kathode resistors:

From top to bottom: Ck compensated for Rk = {1⁄7, 1⁄3, 1, 4, 16} × Zpin. I adapted the lower two values to my suggestion in part three that from Rk = 1⁄20 × Zpin, it does not make much sense to use a bypass cap.

Together, these two look like this:

This sets up a grid for the 15 combined response curves:

I think this looks fascinating; it could make a trainspotter out of any tube nut. We see that on the right the kathode cap is doing all the work, a decade or more to the left the B+ cap takes over. At no moment the curves are steeper than first order.

Here is the same graph, zoomed-in on the top-right area, where your frequency of interest is:

What you see here is what determines what it is going to sound like.

Here is where I feel I have taken control of the issue. For single- and two-capacitor decoupling of resistance-loaded triode gain stages there are straightforward strategies and calculations for setting them up. And with that, it has become possible to plan backwards, i.e. if you want to end up with a certain value-range of capacitors, what architectural, tube and loading choices are available to you. For instance, if your goal is < 10µF kathode decoupling caps, then you now know that low-gm tubes is your game.

tinker, tinker

A word of caution. Some people investigate operating points of tubes by changing kathode and/or loading resistors, and then evaluating (measuring, listening) the result. Some people experiment by plugging a different type of tube into an existing circuit (no, not tube rolling; really a different tube type, with different characteristics).

I have nothing against this form of empirical research. But I urge you to check how much your bandwidth has moved, through changes in Rk, Rload and/or µ & rp. Remember, the lower and higher limit of your bandwidth move in concert. I suspect that often the resulting change will be innocent, but (anywhere near) half-an-octave of bandwidth movement (-30% or +40%) is significant—then you may ask yourself which changes you are listening to.

Adapt your capacitor value(s) when changes in Rk , Rload and/or µ & rp have significant impact on your bandwidth.

crossover success

The reason all graphs above (except the last one) take such a wide-ranging view (17 octaves and 40dB) is that I wanted to investigate what Herb Reichert described in sound practices issue 5: in a multi-amped system, use the time constants of a power amp to implement the crossover of its driver.

Say you got a higher-midrange horn in your system that needs to be integrated with a 500Hz 3rd-order high-pass, could the time constant of gain-stage decoupling be used for that, together with two others (e.g. from coupling caps)? I expect such a solution to be a hippie’s dream, sounding so free and liberated to be intoxicating (‘do not partake in traffic, or operate heavy machinery, for eight hours after listening to it’).

Checking the graph (repeated from above), I say that the responses shown are not fit for crossover use. The three curves that look remotely promising (bottom horizontal strain) are the ones for a humongous kathode resistor. A much more regular situation is the strain for Rk = Zpin, shelving here at -8dB.

So instead, I tried something else:

Same selection of 5 different kathode resistors, but now looks the part, doesn’t it? What did I do? I set C to 1.05 × Cideal; Cdk then automatically balloons to 21 × Ck. Thinking about that rationally, I say that if you are going to do the effort to get to 105% of Cideal, you might as well go for 100% and use fixed bias, omitting what is for sure a very large-value kathode capacitor.

My conclusion from this is that a time constant of a triode gain stage can be used for crossover duty, but only by using fixed bias and Cideal.

cookbook

Ah, you don’t want to read so much, you just want to cook? That’s OK, just follow this recipe.

ingredients


You take a resistance-loaded triode gain stage (no, not OPT-loaded), its B+ decoupled with a resistor and biassed by a kathode resistor.

preparation

To calculate the B+ capacitor for your ƒ-3dB—

1. freely pick the value of Rfilter for the voltage drop you need for your gain stage, resulting in nice voltage headroom, good linearity and no scorching voltage on the plate.

2. calculate the B+ capacitor, set ƒfilter 5 (hippie) to 10 (Nixon) times lower than your ƒ-3dB:
C = 1 / [2πRfilterƒfilter]
That was simple; filter considerations simply overruled everything else. To calculate the right kathode capacitor for your ƒ-3dB—

3. calculate what the problem looks like at the kathode, Zpin:
Zpin = (rp + Rload) / (µ+1)
4. calculate the capacitor to decouple this at your ƒ-3dB, Cpin:
Cpin = 1 / [2πZpinƒ-3dB]
5. compensate the pin capacitor for the kathode resistor:
Ck = √(1 + 2Zpin/Rk) × Cpin
6. calculate the ideal capacitor value (and shed a tear when you see how small it is):
Cideal = 1 / [2π(rp + Rload)ƒ-3dB]
7. calculate the definitive kathode capacitor, taking the B+ capacitor into account:
Cdk = Ck / [1 - Cideal/C]
8. Enjoy.

here is a couple I made earlier

Now for some examples. They are all from real-life circuits from the pages of sound practices, with where necessary a slight adaptation. The ƒ-3dB I gleaned from the coupling cap(s) in the circuit. The stiffness of the B+ filter I had to set myself, going by my intuition.

example 1: a 6sn7 (rp = 7k, µ = 20) stage; Rload = 24k, Rk = 1k; ƒ-3dB = 9Hz.

To calculate the B+ capacitor—

1. freely pick the value of Rfilter. It is 6k2 in the circuit.

2. calculate the B+ capacitor, I set ƒfilter 7 times lower than 9Hz:
C = 1 / [2πRfilterƒfilter] = 1 / [2π × 6k2 × 9/7] = 20µF
To calculate the right kathode capacitor—

3. calculate what the problem looks like at the kathode, Zpin:
Zpin = (rp + Rload) / (µ+1) = (7k + 24k) / 21 = 1k48
4. calculate the capacitor to decouple this at your ƒ-3dB, Cpin:
Cpin = 1 / [2πZpinƒ-3dB] = 1 / [2π × 1k48 × 9] = 12µF
5. compensate the pin capacitor for the kathode resistor:
Ck = √(1 + 2Zpin/Rk) × Cpin = √(1 + 2 × 1k48/1k) × 12µF = 24µF
6. calculate the ideal capacitor value (sniff):
Cideal = 1 / [2π(rp + Rload)ƒ-3dB] = 1 / [2π × (7k + 24k) × 9] = 570nF
7. calculate the definitive kathode capacitor, taking the B+ capacitor into account:
Cdk = Ck / [1 - Cideal/C] = 24µF / [1 - 570nF/20µF] = 25µF
The B+ capacitor is 35 times larger than Cideal, so there is only a slight adjustment to Cdk.

example 2: a 5687 (rp = 2k5, µ = 17.5) stage; Rload = 14k, Rk = 82Ω; ƒ-3dB = 3.3Hz.
  1. Rfilter is 2k5 in the circuit.
  2. I set ƒfilter 10 times lower than 3.3Hz:
    C = 1 / [2πRfilterƒfilter] = 1 / [2π × 2k5 × 3.3/10] = 194µF
  3. Zpin = (rp + Rload) / (µ+1) = (2k5 + 14k) / 18.5 = 892
  4. Cpin = 1 / [2πZpinƒ-3dB] = 1 / [2π × 892 × 3.3] = 54µF
  5. Ck = √(1 + 2Zpin/Rk) × Cpin = √(1 + 2 × 892/82) × 54µF = 258µF
  6. Cideal = 1 / [2π(rp + Rload)ƒ-3dB] = 1 / [2π × (2k5 + 14k) × 3.3] = 2.9µF
  7. Cdk = Ck / [1 - Cideal/C] = 258µF / [1 - 2.9µF/194µF] = 262µF
A low ƒ-3dB and a very small Rk (equal to Zpin/11) conspire here to produce large cap values.

example 3: another 5687 stage, but this time it is direct-coupled to the stage before it, resulting in a humongous kathode resistor; Rk = 19k5, Rload = 50k; ƒ-3dB = 3Hz.
  1. Rfilter is 1k in the circuit.
  2. I set ƒfilter 10 times lower than 3Hz:
    C = 1 / [2πRfilterƒfilter] = 1 / [2π × 1k × 3/5] = 265µF
  3. Zpin = (rp + Rload) / (µ+1) = (2k5 + 50k) / 18.5 = 2k8
  4. Cpin = 1 / [2πZpinƒ-3dB] = 1 / [2π × 2k8 × 3] = 19µF
  5. Ck = √(1 + 2Zpin/Rk) × Cpin = √(1 + 2 × 2k8/19k5) × 19µF = 20µF
  6. Cideal = 1 / [2π(rp + Rload)ƒ-3dB] = 1 / [2π × (2k5 + 50k) × 3] = 1µF
  7. Cdk = Ck / [1 - Cideal/C] = 20µF / [1 - 1µF/265µF] = 20µF
Because of the humongous kathode resistor and the large B+ cap, Cdk is very close to Cpin.

example 4: and now for the real deal. The example above was adapted from the second stage of Kondo’s ongaku, but that stage actually has one extra kink:

There is an unbypassed 1k resistor in the kathode circuit. This changes the characteristic of the tube; a rp = 2k5, µ = 17.5 tube becomes a rp = 21k, µ = 17.5 one. rp is up by a factor 8.4, gm is reduced by the same factor. The other parameters are as before: Rk = 19k5, Rload = 50k; ƒ-3dB = 3Hz.
  1. Rfilter is unchanged (1k).
  2. As is the B+ capacitor (265µF), but we can now see step by step how the kathode capacitor calculation changes—
  3. Zpin = (rp + Rload) / (µ+1) = (21k + 50k) / 18.5 = 3k8
  4. Cpin = 1 / [2πZpinƒ-3dB] = 1 / [2π × 3k8 × 3] = 14µF
  5. Ck = √(1 + 2Zpin/Rk) × Cpin = √(1 + 2 × 3k8/19k5) × 14µF = 16.5µF
  6. Cideal = 1 / [2π(rp + Rload)ƒ-3dB] = 1 / [2π × (21 + 50k) × 3] = 747nF
  7. Cdk = Ck / [1 - Cideal/C] = 16.5µF / [1 - 747nF/265µF] = 16.5µF
That is disappointing, I thought we would be up for a big change here, with gm plummeting by 8.4. But from part 3 we know that Cpin ~ gm / [1 + Rload/rp], and rp also changes a lot, so [1 + Rload/rp] compensates by changing by a factor 6.2, leaving little net change.

example 5: everybody’s friend, an ecc83 (rp = 65k, µ = 102) stage; Rload = 220k, Rk = 2k2; ƒ-3dB = 10Hz.
  1. Rfilter is 4k7 in the circuit.
  2. I set ƒfilter 5 times lower than 10Hz:
    C = 1 / [2πRfilterƒfilter] = 1 / [2π × 4k7 × 10/5] = 17µF
  3. Zpin = (rp + Rload) / (µ+1) = (65k + 220k) / 103 = 2k77
  4. Cpin = 1 / [2πZpinƒ-3dB] = 1 / [2π × 2k77 × 10] = 5.8µF
  5. Ck = √(1 + 2Zpin/Rk) × Cpin = √(1 + 2 × 2k77/2k2) × 5.8µF = 11µF
  6. Cideal = 1 / [2π(rp + Rload)ƒ-3dB] = 1 / [2π × (65k + 220k) × 10] = 56nF
  7. Cdk = Ck / [1 - Cideal/C] = 11µF / [1 - 56nF/17µF] = 11µF
Yes, you could have used the ideal capacitor value of 56nF to decouple this stage. With the B+ capacitor being 330 times larger than that, it was not worth the trouble to calculate the correction to Cdk.

doggy bag

Here are the take-home points of this instalment—
  • The B+ and kathode capacitors work in series, resulting in a first-order roll-off.
  • in most practical circuits the value of the B+ capacitor is way too large, and it is the kathode capacitor that controls the time constant of the stage.
  • For crossover duty, use fixed bias and Cideal for the B+.
  • Calculating the two capacitor values is a straightforward, cookbook procedure, where you are completely free to make all the design decisions (tube type, bias, loading, voltages).
On top of that there are the take-home points for this four-part series—
  • One cap—decoupling either B+ or kathode—or two? After picking any of these three topologies, there is a straightforward path to achieving what you need.
  • Using only a B+ cap (and no Ck) is by far the most difficult route of the three. Either you work real hard, or you will end up with a humongous time constant.
  • With a kathode capacitor the time constant of the stage can be always be reduced and easily controlled.
  • Achieving Cideal is the holy grail. It is the smallest possible cap value that implements your desired time constant, and boy is it a lot smaller than what we are used to—see the cookbook above and the examples in part two.
  • Only the, by far, most difficult route leads to the holy grail.

is this the end?

Seeing the reactions to this series as it was published, I have to make something clear. It was about conventional cap decoupling of resistance-loaded triode gain stages. As noted before, this topology has been used in an ungodly amount of circuits in the past and it will still be the tube circuit industry standard in the future.

This series was not about—
  • tricky cap decoupling (e.g. ultrapath);
  • choke loading, nor transformer loading;
  • tetrodes, nor pentodes;
  • triodes stacked in tricky ways; e.g. cascode, SRPP, mu-stage, SEPP;
  • push-pull, nor differential;
  • power output stages, nor kathode followers.
It was not about any of these. But I am interested in a majority of these, which means I want to understand them—and then write it up—to the level I achieved in these first four instalments of this series.

Yes, this series will continue, stay tuned.
at

1 comment:

  1. John Day7/1/16 16:05

    Thanks Peter,
    The topology where the B+ cap and the cathode cap meet at the cathode is particularly interesting. I can't do the math... sorry

    ReplyDelete