Two cathode resistor bypass questions

[Sherman]

I've been trying to understand cathode resistor bypass a little better. Specifically- how to calculate the value of the bypass cap and the effect of bypassing a larger electrolytic with a smaller quality cap.

I have the formula-
1/(2*pi*f*Rk) (where "f" is the lower frequency, usually 20).

I've also seen it written as-

1/(2*pi*20*1/gm||Rk)

However my answers don't seem to match up with calculated values if I use TubeCAD (or with any other values ).

For example TubeCAD gives a value of 126uF for the cap on a 6DJ8 with a cathode resistor of 205 ohms. Doing the math I get-

1/(2*3.1416*20*205) or 1/25,761.12 giving 0.000038818188 which is only 38.8uF.

What am I doing wrong?

Second, once I get this part down is there a formula for calculating the value of say a film and foil cap to bypass the 'lytic?

Thanks for any help!

[Geek]

The formula I've used with no trouble is use a capacitor equal in reactance to the value of the cathode resistor, then double the value. So, if your math says 4.7uF at 20Hz = Xc = Rk, make it 10uF.

And if you use a low ESR, non-inductive electrolytic, like ones for SMPS, the need for a film/foil cap around it is practically eliminated. Only in the most critical of applications have I heard a sonic difference (OTL's to be specific). I just use an arbitrary value of 1/10 the value of the electrolytic.

[i_should_coco]

Actually, the frequency, f, is the -3dB point. You will in all likelihood want a -3dB point well below 20Hz to minimize phase shift in the bass region. A common value is 5Hz. This will account for most of the discrepancy.

In addition, it's necessary to take into account the impedance looking into the cathode which is often neglected when calculating the bypass cap. The cathode resistance is calcluated by Rk' = (Rl+Ra)/(mu+1) where Ra is the load resistance. So please note that the anode resistor affects this. This is in parallel with the cathode resistor.

So, if we assume 5Hz as our -3dB frequencey and let's say 10k load, mu for 6DJ8 is 33 & Ra is 2.6k giving us a cathode resistance of

(10k+2.6k)/(33+1) = 370R. In parallel with 205R cathode resistor gives us 122R.

Calculating capacitor using 1/(2.pi.F.C) gives us 241uF. Of course this is wildy different form the results you get above! Mainly I suspect because I'm guessing the frequency for -3dB.

In practice, the value isn't *that* critical as long as it's big enough.

Cheers,
Pete

[Sherman]

Peter,

Thanks for reply!

Originally posted by i_should_coco
[B]Actually, the frequency, f, is the -3dB point...

In one of the explanations I found online the write said that using a value of 20 in the formula actually resulted in value that would give a -3dB point of between 4Hz and 5Hz. I didn't understand his explanation for that but I've seen a couple other places where 20 is used so I went with it. I'll try it for different values.

...(10k+2.6k)/(33+1) = 370R. In parallel with 205R cathode resistor gives us 122R.

That was a very clear explanation! Thanks.

In practice, the value isn't *that* critical as long as it's big enough.

That's good news. So far I've been using TubeCAD and SEAmp CAD to play with various tube circuits but want to understand more about how to calculate these values. I can use loadlines, calculate the cathode resistor and anode resistor, current through the tube and more but my use of this particular formula never matched the calculated values.

[Sherman]

Originally posted by Geek
...

Only in the most critical of applications have I heard a sonic difference (OTL's to be specific). I just use an arbitrary value of 1/10 the value of the electrolytic.

That's what I like, something nice and easy to figure out! Even I can divide by 10.

[i_should_coco]

Hello Sherman,

Originally posted by Sherman
In one of the explanations I found online the write said that using a value of 20 in the formula actually resulted in value that would give a -3dB point of between 4Hz and 5Hz. I didn't understand his explanation for that but I've seen a couple other places where 20 is used so I went with it. I'll try it for different values.

Just to clarify, the 3dB point is where your gain is down to 50% of it's orignal value. The cathode bypass & cathode resistance act like a potential divider, so the 50% gain point is when the two resistances are equal. It's possible that the explanations you found have some "correction" factor to account for this. Not sure though.

That's good news. So far I've been using TubeCAD and SEAmp CAD to play with various tube circuits but want to understand more about how to calculate these values. I can use loadlines, calculate the cathode resistor and anode resistor, current through the tube and more but my use of this particular formula never matched the calculated values.

As you've found, there are a few different ways folks use to calculate this, unfortunatelt we don't know exactly what formulas the CAD applications use to derive these. Do the two applications agree on the value, or are they different too?

BTW, I'm not a huge fan of bypassing, but one rule of thumb (from Morgan Jones) is to use a cap. that is 1/100th of the value.

Cheers,
Pete

[kevinkr]

By convention the -3dB point is referred to half power and not half voltage. In the world of voltage this value corresponds to about 0.707 which when you do the math will equal half power.

So for voltage gain a value of 0.707 of the nominal numerical gain value is equivalent to the -3dB response point.

A 4 - 5Hz -3dB point is a very reasonable choice to avoid excessive phase shift at 20Hz and above.

[i_should_coco]

Oops! Sorry, yes, for voltage gain 50% would be -6dB.


Cheers,
Pete

[Giaime]

Appendix B: The math behind the equivalent cathode impedance:

2Equivalent cathode resistance determination:
If you inject a voltage of 1V at the grid, with the cathode grounded, you get an equivalent "internal" voltage source of mu*Vgk, which then produces a current of mu*Vgk/ra (this is effectively a transconductance - the voltage at the grid produces a corresponding current change in the plate circuit). There is no current flowing in the grid circuit, so the 1V at the grid contributes no direct current of its own except through the transconductance of the tube.

However, if you inject a voltage of Vgk at the cathode, with the grid grounded, you get a current of mu*Vgk/ra plus the current due to Vgk injected at the cathode, which now appears directly across (ra+Rp), since they are in series. This means the equivalent voltage source is now mu*Vgk + Vgk, which is equal to (mu+1)*Vgk, so the current produced by this source is (mu+1)*Vgk/(ra+Rp).

This gives an impedance, seen "looking into" the cathode, of:

Vgk/((mu+1)*Vgk/(ra+Rp))

which is equal to:

(ra+Rp)/(mu+1)

since the Vgk terms cancel.

Example:
With a plate resistor of 100K, and internal plate resistance of 62.5K and a mu of 100, the 12AX7 will have a cathode impedance of (62.5K+100K)/(100+1) = 1609 ohms. This impedance is then in parallel with the actual cathode resistor, so if you used an 820 ohm cathode resistor, the actual cathode resistance would be 1609||820 = 543 ohms. This is the value you would use when calculating frequency response, so if you had a 0.68uF cap bypassing the 820 ohm, you'd end up with an actual -3dB point of 1/(2*pi*0.68uF*543ohms) = 431Hz.

from http://www.aikenamps.com/CommonCathode.htm
very good explained

[Sherman]

Originally posted by Giaime


from http://www.aikenamps.com/CommonCathode.htm
very good explained

Giame,

Great link. Thanks. I've been putting together a spreadsheet that will run on my Palm to help calculate various bits for tube amps. I work all the formulas manually and try to understand them before plugging them into the spreadsheet.

I can now get repeatable results that I understand but I still can't get the same results as TubeCAD. The program help file gives the formula for cathode resistor bypass cap as-
1/(2*pi*20*Rk*(1/Gm)). He is using 20Hz as the low frequency cutoff (not sure if that is the same as the -3dB point to him or not).

At this point I'm comfortable with the values I'm getting with the help of this thread so I won't worry about matching TubeCAD!