cap decoupling /3⅓ : proof

This is an addendum to part 3 of the lecture that I held at the 2015 European Triode Festival (ETF), where I introduced a correct formula for decoupling the kathode of resistance-loaded triode gain stages. If you are not familiar with it, then you really want to read that first.

Here I will present what several people have asked me for: a derivation of this formula. Up to now the whole thing was based on empirical observation; can it also be proven?

To take a swing at deriving the formula, I had to brush up my complex math skills; it has been 25 years since I last used them in university. Luckily, pages with basic math definitions exist (thanks!) so I was good to go.

setting up

(x: frequency in octaves; y: decibels)

What was the goal again? Ah yeah, it was to make the black curves match the orange one, for as long as that is practically possible. The orange curve is the ideal of C_pin decoupling Z_pin (without taking the kathode resistor into account). The black curves are the same C_pin and Z_pin with different R_k values thrown into the mix. See part three for all the details.

To achieve the goal, we look at one frequency of interest (encircled above) that is above ƒ_-3dB, take the value of the orange curve at that point and figure out how much the kathode capacitance has to be increased to make each black curve match. The formula for that is:

|1 / [1 + 1/j2πC_pinƒ_interest]| =
|1 / [1 + 1/(j2πyC_pinƒ_interest + Z_pin/R_k)]|

y is here the factor with which C_pin has to be increased to make the match. The formula is already normalised for the actual loading of the tube by R_load; R_k is expressed in terms of Z_pin. We make this look slightly less unappetising by defining—

D = 2πC_pinƒ_interest

and by dropping the one-over part on both sides:

|1 + 1/jD]| = |1 + 1/(jyD + Z_pin/R_k)|

I have called it D for distance—the one between ƒ_-3dB and ƒ_interest. Above is the (implicit) equation I had my math grapher plot in part three:

(x: log₂(R_k/Z_pin); y: log₂(y))

This is for different values of D: D=1 is the right-most, this means ƒ_interest = ƒ_-3dB. The left-most curve is for D=10, i.e. ƒ_interest is one decade higher than ƒ_-3dB. We can observe here (just by watching):

1. each curve collapses on the left; this is where the matching we are trying to achieve is no longer practical—the shelving of the response interferes;
2. every high-D curve incorporates the good parts of the low-D curves;
3. to get good coverage of a practical range of kathode resistors (compared to Z_pin), we need to work with a D that is 10 or higher.

So this creates a rule we will work with here:

D > 10

Before we get going there is one more convenience definition—

x = Z_pin/R_k

This makes our working equation something I can handle:

|1 + 1/jD]| = |1 + 1/(jyD + x)|

proof

The job is now to solve for y. First untangle the devision on the right by multiplying top and bottom with the conjugate of the denominator:

|1 + 1/jD]| = |1 + x/(x² + y²D²) - jyD/(x² + y²D²)|

Then take the magnitude on both sides (drop the square roots immediately):

1 + 1/D² =
1 + 2x/(x² + y²D²) + x²/(x² + y²D²)² + y²D²/(x² + y²D²)²

Drop the ones on both sides and multiply all terms by x² + y²D²:

y² + x²/D² = 2x + (x² + y²D²)/(x² + y²D²) = 2x + 1

Rearranging that:

y² = 1 + 2x - x²/D²

Now we remember that D > 10. That means that even for extremely small-value kathode resistors (R_k = Z_pin/20, i.e. x=20) 2x will be at least 10 times larger than x²/D². For the more normal value of x=1, 2x is 200 times larger than x²/D². So now we perform the trick that no pro-physics proof shall be without: drop the piddly term to get a nicer result:

y² = 1 + 2x

Filling in x and performing square root on both sides:

y = √(1 + 2Z_pin/R_k)

There we are, with y being the factor to multiply C_pin with to get the R_k-compensated C_k, we get:

C_k = √(1 + 2Z_pin/R_k) × C_pin

As they say in physics: bingo.

If you thought it a bit fast to drop the x²/D² term just like that, remember that to investigate really small-valued kathode resistors, you need a really large D. In the second graph above, the x²/D² term is what brings on the collapse for every curve. To properly investigate, a collapse has to be avoided, i.e. D has to be large enough so that x²/D² is insignificant.

happy ending

It is now proven, every kathode resistor can be correctly bypassed, with smooth, predictable results:

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