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!

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]

...

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]

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!

 

[Giaime]

I just saw in the program help, and Sherman the formula isn't that but this:

C = 1/(2·pi·20·[1/Gm || Rk]) which means that 1/gm is paralleled to Rk, not multiplied.

Which to me doesn't appear to be correct. For example using the formula above will lead (6DJ8 example) a cutoff that is at 10Hz, not 20Hz (for cutoff I mean the -3dB half power point, 0.707times the midband gain).

I did the math and that formula seems to be very different from reality, since it supposes that 1/Gm = (ra+Rp)/(mu+1). This could be only if the Gm is not the "datasheet one" (11mA/V for the 6DJ8) but the actual one calculated in that circuit, and if I am not wrong, it is exactly that.

I still suggest to use the old style formula that is described in the link I gave some posts ago.

 

[i_should_coco]

Hi Giane, Sherman,

I did the math and that formula seems to be very different from reality, since it supposes that 1/Gm = (ra+Rp)/(mu+1). This could be only if the Gm is not the "datasheet one" (11mA/V for the 6DJ8) but the actual one calculated in that circuit, and if I am not wrong, it is exactly that.

I think using the actual in circuit value for gM is probably a good thing? gM will vary somewhat with operating conditions and usually what is achievable in circuit is lower that the datasheet (they tend to be optimistic).

I still suggest to use the old style formula that is described in the link I gave some posts ago.

That what I use, but I guess either is acceptable, like I say, I don't think it's that critical.

Cheers,
Pete

 

[Sherman]

Giaime,
Thanks for delving into that. I also reread the help and found that I had mistakenly written it as multiplication when it should be parallel.

...

I did the math and that formula seems to be very different from reality, since it supposes that 1/Gm = (ra+Rp)/(mu+1). This could be only if the Gm is not the "datasheet one" (11mA/V for the 6DJ8) but the actual one calculated in that circuit, and if I am not wrong, it is exactly that.

OK, then it isn't just me! I couldn't get his calculated value no matter what I did. Though I didn't recalc Gm for the specific circuit but just used the datasheet value.

I still suggest to use the old style formula that is described in the link I gave some posts ago.

Definitely! I've created a pretty good little spreadsheet using the "old style" formula and have double-checked it against some actual values in real schematics and feel comfortable that it is close enough.

Thanks again for your help in this.

Pete-
That is what I'm thinking- using the "old" formulas gets me close enough to what is after all a non-critical value.

 

[Giaime]

close enough to what is after all a non-critical value.

Sorry people, but after all I don't think that's a non-critical value. I extensively use proper dimensioning of cathode bypass capacitor to define low frequency response of a circuit. For example, the relatioship between the low frequency cutoff of an output transformer and this kind of cutoff can be employed for our needs: just as an example, usually for inexpensive SE triode output stages I tend to place a 40Hz -3dB point. It can seem too much: but please note that the output transformer won't pass 40Hz as well, it will only saturate increasing distortion. High -3dB point in SE amps helps with low frequency distortion.

This is just an example, another example would be the infrasonic response of phono preamps (a topic that could take books).

And for guitar amps, tone shaping with cathode bypass capacitor is EVERYTHING!!!

And you should consider the tradeoff between film/foil (or paper in oil) caps and electrolytics. If I can determine PRECISELY the value I need, I can choose wheter a PIO or an electrolytic can be employed. If I don't know precisely the value, I will probably set for an electrolytic (thus ruining the sound) where maybe a PIO would have been totally acceptable.

So, as any other electronics law, it's not uncritical when you want to have CONTROL on the whole amp behaviour and response.

 

[rdf]

Hi Giame. I tried the same technique with a low power, no NFB single-ended. As the OPT power rating was ridiculously conservative I went for LF extension instead. Spice analysis and measurements into a resistive load both showed dead flat response at 1 watt to below 10 Hz. However into the rising low frequency impedance of a real speaker it was cartoon bass time. Additional measurements back on the bench revealed a damping factor which fell dramatically with decreasing frequency, below 1 at the very bottom. My solution was a conversion to fixed bias and using the OPT secondary as a cathode load.

 

[rdf]

By complete coincidence while organizing files today I came across the measurements related to the above. I forgot taking them! The pdf contains voltage measurements at the secondary winding for a no-NFB SE amp at various frequencies, with and without 8 ohm termination. An Openoffice spreadsheet was used to derive Zout and damping factor. The output tube is cathode biased and the value of the capacitor selected to 'tailor' the low frequency response in the 20 Hz range. Unfortunately I didn't record values but recall them as somewhere in the range of ~300 ohms (I like the EL84 high and cold) and 33 uF. The RC time constant falls into the low audio range. As mentioned above, the values were chosen to optimize low frequency extension into an 8 ohm resistor. The 'FR resp' column next to '8 ohms' shows a response less than 0.1 dB down at 20 Hz.

The effect was immediately obvious and audible on a JX92s in a short MLTL tuned for the low 40 Hz range.

 

[Giaime]

May I suggest you, for the next time, to take measurements with the actual speaker as load? Far better results because at those frequencies it's difficult that your speaker represent an 8ohm load.

 

[rdf]

It's all about circuits and choices. Into a speaker the low frequency rise of this no-NFB EL84 SE would be reduced compared to a complete open circuit. In the case of the MLTL signifcantly, though now with two peaks to follow the alignment's double-hump impedance curve. I can also see how reducing the capacitor bypass further still could tame some of that remnant. However, the amp's Zout isn't dependent on load (the ratio DF is) and reducing the bypass cap will increase the stage output impedance to higher frequencies still. As I mentioned, a small bypass cap was very audible even with the 5" Jordan. It's one factor to consider when optimizing the for best tube/OPT/speaker match. For a tube with the sensitivity of an EL84 and an OPT without cathode feedback windings, secondary cathode feedback allows a cake and icing lunch: improved extension, lower distortion, better damping factor. A 2A3 or 45 is a completely different story.