cap decoupling of triodes /3

This is part 3 (out of 4) 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’, ƒ_interest, 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. And if the term C_ideal does not sound familiar, (re)read part two.

In this instalment we will look at kathode decoupling in isolation—

Note that the B+ is regulated. Any regulator will do, as long as its impedance is low and resistive in the ƒ_-3dB region. [correction: a cap multiplier also works, because it ain’t nothing but a huge, amplified capacitor.]

In today’s instalment, the game is really similar to the B+ decoupling in part 2: decoupling of the tube’s rp and load. This time, the kathode capacitor (C_k) has to do the job. And the kathode resistor (R_k), you ask? It will turn up later in this story, making the job a little, or a lot, harder.

At this point is is super convenient to introduce the pin impedance:

Z_pin is exactly what the ‘whole problem’ looks like, when you are ‘down there’ in the kathode circuit and look straight up the kathode pin of the tube. It is defined as—

Z_pin = (rp + R_load) / (µ+1)

You see it consists of the familiar rp + R_load, and everything is µ+1 times more challenging at the kathode. I picked the name Z_pin because it is snappy and in no way can be confused with ‘kathode resistor.’ It will be easier for the both of us.

What is super convenient for us is that Z_pin encapsulates the specifics of the tube (rp and µ) and its loading (R_load/rp). From now on, by relating everything to Zpin, every conclusion is valid for any triode tube and any kind of resistive loading.

caps, no caps

Before we get to kathode caps, here is a way of avoiding one. Lately, some of my friends have been a fan of biassing tubes in the following way:

From a clean low-voltage supply a relatively large current is fed to a small-value kathode resistor. R_k is so small that no cap is required. How small a value should that be? May I suggest Z_pin/20? It’s OK if you feel more comfortable with a different fraction of Z_pin, but I will use this value from now for ‘where is stops to make sense’ to use a kathode cap at all.

back at the ranch

Now we take the first step in kathode decoupling. Z_pin is fully decoupled at your own ƒ_-3dB by the pin capacitor:

C_pin = 1 / [2πZ_pinƒ_-3dB]

This all is very analogue to C_ideal and also here I can say that it will never get better/lower than this. I will now go through the same examples as in part two, to give you a feel for C_pin—again I rolled the dice for a ƒ_-3dB and… all of them are @12Hz.

6sn7, R_load = 24k, C_pin = 9µF. Yep, it does not take so much to decouple this general-purpose driver tube.

26, R_load = 24k, C_pin = 4µF. Very appetising, no? With its lower µ it is not the same story as the 6sn7—as it was for the B+ in part 2.

ecc83, R_load = 100k, C_pin = 8.5µF. The high µ gets this high-rp tube in the same class as the 6sn7.

ecc88, R_load = 10k, C_pin = 36µF. A step up, but no longer a factor twelve larger.

ec8020, R_load = 2k2, C_pin = 238µF. Wow! Remember it does not get lower than this. At 3Hz, the pin capacitor is already close to a milliFarad.

el34(triode), R_load = 2k2, C_pin = 49µF. Same rp, but way lower µ than an ec8020.

In general we can observe that C_pin ~ gm / [1 + R_load/rp]. That is, the pin capacitor is proportional to the gm of the tube, regardless of rp and µ. The racier the tube, the larger is C_pin. Besides that, using larger plate resistors lowers the value of C_pin.

the riddle

Now we add the kathode resistor to the mix:

(x: frequency in octaves; y: decibels)

The orange line is the decoupling achieved by C_pin on its own. The black line shows the response of this C_pin and a kathode resistor that is equal to Z_pin. What the arrow shows is that the onset of the roll-off has moved to the right, by what looks like three-quarters of an octave. That is, the time constant moved by that much.

Here we see what happens when the kathode resistor becomes smaller than the pin impedance:

On the left we see that the smaller the kathode resistor, the smaller the final attenuation is. On the right we see that the smaller R_k, the more the time constant is reduced. ‘But the attenuation is only a dB or two!’ you say. Yes, but it forms a shelving equalisation and those are super easy to hear as change in tonal balance,

Here is what happens when the kathode resistor gets larger than the pin impedance:

We see that the higher R_k, the closer we get to a C_pin roll-off.

Here are all five values of Rk in one graph:

And here I have zoomed in on what I call the problem zone:

This is the area where your frequency of interest is most likely to fall. What the problem is? Any occurrence of a black line below the orange one. We need to find the minimum we have to do to the kathode capacitor to get all the black ones to track the orange line as long as possible.

Regular readers of this blog may have noticed that for a long time I have been dissatisfied with the fact that articles and books have been either vague (‘just put enough’) or plainly wrong (‘just calculate a cap with R_k and ƒ_-3dB’) on kathode capacitors. In the spring of this year I decided to investigate.

machine learning

My first approach was to let the computer solve it. I looked at the problem zone—

and said: say that vertical line is the frequency of interest. Can my trusty math grapher show me how much the kathode capacitor has to be increased to get each of the black lines exactly on top of the orange one at that point?

Yes it can:

(x: R_k compared to Z_pin, 0 is equal, doubling to the right [1=2×, 2=4×…], halving to the left [-1=½, -2=¼…]—technically log₂(R_k/Z_pin);
y: C_k compared to C_pin, 0 is equal, doubling per tick [1=2×, 2=4×…]—technically log₂(C_k/C_pin))

The reason there are several graphs is that because we are acting at the frequency of interest, it really matters how much lower ƒ_-3dB was set:

First observations are that on the right we can see that no matter how large the kathode resistor becomes (R_k = 256Z_pin on the far-right), the needed kathode capacitor never falls below C_pin. On the left we see that the lower the ƒ_-3dB is compared to ƒ_interest, the larger the range of R_k that can be graphed.

This graph is what I had to show you in spring. Normally, I would now explain how you would use this nomograph for your designs. But all of it is superseded, because something cool happened.

getting lucky

One morning, I could not take it any longer. Why is there no straightforward formula for C_k? As I observed in part one of this series, this is the journeyman level of tube design. Triode gain stage 101 should go like this: 1) select tube and load for gain; 2) determine and set up biassing; 3) calculate caps with ƒ_-3dB; done.

So I took a different approach. I thought ‘let’s determine the correct C_k that fully compensates for R_k by hand—for a couple of R_k values. And then see if I can deduct a formula from all this.’

I bisected the increase in kathode capacitor value of the with-R_k curve (black), until I got the best match with the C_pin roll-off (orange):

I started with R_k = Z_pin. I determined that C_k had to be 1.732 × C_pin (pictured above). Now that number looked familiar to me; something from trigonometry in school. So I put it in my calculator and pressed the x² button. Ah, the increase is √3.

Then I set R_k to Z_pin/3. I determined that C_k had to be 2.645 × C_pin. Immediately I squared the number. It is √7.

Finally I set R_k to 3Z_pin. I determined that C_k had to be 1.291 × C_pin. The increase is √1⅔.

That was enough to deduct that—

C_k = √(1 + 2Z_pin/R_k) × C_pin

can I kick it?

So I had a formula, but is it any good? I thought I’d see if it has any correlation with that computer-generated graph I showed you just moments ago:

The orange curve is the formula. That is encouraging; I think I am on to something, here. The blue vertical line marks where R_k = Z_pin/20; i.e. where it stops to make sense to use a kathode cap at all. At that extreme position, C_k = 6.4 × C_pin.

So that is what good correlation looks like. And now some examples of what does not correlate. First up:

The blue curve is for the formula that you can find, for instance, in Morgan Jones’ book. It takes R_k in parallel with Z_pin and calculates C_k from that for ƒ_-3dB. Rationally, I can completely follow the reasoning that goes into that formula. However, we see how it races away on the left. At the lower extreme (Z_pin/20) the blue line prescribes a kathode cap that is 3.4 times higher than the formula I found.

Here is another non-correlation:

This straight line shows what happens when you simply use R_k and ƒ_-3dB for calculating C_k. Yes, it continues below zero on the right.

problem solved

Remember that close-up of the problem zone I showed you before? Here is it again:

The orange line is the curve of the time constant we are trying to achieve. The black lines are the uncompensated curves of C_pin in combination with R_k = {¹⁄₁₆, ¹⁄₄, 1, 4, 16} × Z_pin.

Before we get it right, here is what getting it wrong looks like:

This is what happens when you simply use R_k and ƒ_-3dB for calculating C_k. The time constants are all over the place; it is worse than the uncompensated situation. It is chaos.

Here the formula to compensate C_k for R_k has been applied:

It looks like only four lines because the one for R_k = 16 × Z_pin is still faithfully tracking the orange line. This is what achieving the desired time constant looks like.

Just for the heck of it, here are two more detailed graphs. First, C_pin uncompensated:

(x: frequency in octaves; y: decibels)

The range of kathode resistors is ¹⁄₁₆ to 4, in increments of ×√2, plus 8 and 16—all times Z_pin. From hippie to Nixon, the frequency of interest will fall between zero and 3.3. Note that the time constant for R_k = ¹⁄₁₆ × Z_pin is 2.5 octaves off.

And this is what C_k, compensated for R_k, looks like:

Lovely.

After all this I conclude with confidence: the formula for the kathode capacitor, compensated for the kathode resistor is:

C_k = √(1 + 2Z_pin/R_k) × C_pin

The era of handwaving and disinformation can come to a close. Texts and books on tube amplifiers can from now on use this simple formula (however, notice the licence of this blog’s content, at the bottom of this page).

update: I have now provided a derivation of the formula.

scraping the barrel

One implication of ‘it does not get better than C_pin’ should be clear by now. In a direct-coupled situation:

where a (relative to Z_pin) humongous kathode resistor of the second stage is used, the kathode cap will be basically C_pin.

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), with a regulated B+ (low impedance at ƒ_-3dB) and biassed by a kathode resistor.

preparation

To calculate the right kathode capacitor for your ƒ_-3dB—

1. calculate what the problem looks like at the kathode, Z_pin:

Z_pin = (rp + R_load) / (µ+1)

2. calculate the capacitor to decouple this at your ƒ_-3dB, C_pin:

C_pin = 1 / [2πZ_pinƒ_-3dB]

3. compensate the pin capacitor for the kathode resistor:

C_k = √(1 + 2Z_pin/R_k) × C_pin

4. Enjoy.

here is one I made earlier

For example, a 6sn7 (rp = 7k, µ = 20) stage; R_load = 24k, R_k = 1k; for ƒ_-3dB we use 9Hz—

1. calculate what the problem looks like at the kathode, Z_pin:

Z_pin = (rp + R_load) / (µ+1) = (7k + 24k) / 21 = 1k48

2. calculate the capacitor to decouple this at your ƒ_-3dB, C_pin:

C_pin = 1 / [2πZ_pinƒ_-3dB] = 1 / [2π × 1k48 × 9] = 12µF

3. compensate the pin capacitor for the kathode resistor:

C_k = √(1 + 2Z_pin/R_k) × C_pin = √(1 + 2×1k48/1k) × 12µF = 24µF

4. Bob’s your uncle.

doggy bag

Here are the take-home points of this instalment—

• The job of the kathode capacitor is the same as that of the B+ capacitor: decoupling the tube rp and load.
• There is a capacitor value (C_pin) just perfect for that and it is proportional to the tube’s gm and loading.
• When the kathode resistor has a very small value (< Z_pin/20), it stops making sense to use a kathode capacitor.
• When not very small (> Z_pin/20), the kathode resistor does not makes things easier for us; it reduces the time constant, sometimes quite severely.
• It is straightforward to calculate the kathode capacitor value that decouples exactly at the desired ƒ_-3dB and is fully compensated for the kathode resistor.

And that’s all. Next up part four, where we will go into combining B+ and kathode capacitors in the same circuit.