Poles, Zeros, and Cathode Bypass Caps
Cathode Bypass Caps
from: kg
Date: 1/28/2002 2:40 PM
Subject: Poles and Zeros and Cathode Bypass Caps
Cathode caps, on the other hand, are shelving and continue to pass low frequencies at the unbypassed level — I think there’s only a 6db attenuation below their knee frequency.
not quite true.
the gain through the stage, when the frequency is low and the cap is out of circuit, is set by Rl/Rk. obviously this can be any ratio, even 1 or less than 1. at a higher frequency, when the cap is completely in circuit, its reactance drops to a low value, and swamps the value of Rk, essentially making Rk = 0 and boosting the gain. so you’re right about the type of boost, but it doesn’t have to be limited to 6dB.. the SLOPE is, but the ultimate value is not.
it’s a bit more exact to call it a pole-zero pair… shelving connotates a type of eq that extends as far as possible to the frequency extremes, as in high shelf, low shelf.
the ZERO is where the response of the circuit increases by 6db/octave. in other words, it “turns up.” this is the point where the cap starts to have an effect, and it is defined by the cathode resistor and the bypass cap value. below this frequency, as i said earlier, the gain is defined by Rl/Rk.
as the frequency is continually increased, the response flattens out to its midband value, the max gain from the stage. this is a POLE, where the reponse “turns down.” since we had a response that was increasing at 6dB/oct, we have essentially cancelled that out, leaving a flat response in its place.
the reason the pole occurs at a higher frequency is that the associated resistance is lower… instead of just the Rk and Ck being involved, one must also take into account the parallel effect of the internal cathode resistance itself, estimated as 1/gm. since this will result in a smaller RC time constant, the frequency is correspondingly higher.
for the region of frequencies between the zero and the pole the response is indeed increasing at 6 dB/oct, which is where you got that number stuck in your head. the ultimate gain can be greater or less than than 6dB depending on how many octaves are between the pole and zero.
hth
ken
From: Dave H. @
Date: 1/28/2002 4:50 PM
Subject: Re: Poles and Zeros and Cathode Bypass Caps
for the region of frequencies between the zero and the pole the response is indeed increasing at 6 dB/oct, which is where you got that number stuck in your head. the ultimate gain can be greater or less than 6dB depending on how many octaves are between the pole and zero.
Ken,
If you add a cap across the Rk for a circuit with the usual 1k5 Rk and 100k Rp values the gain at higher frequencies is doubled i.e an increase of 6dB. I think the 6dB number comes from there not from the 6dB/oct slope of the RC. As the frequency is reduced the pole created by Ck/Rk (and the resistance looking into cathode) causes the gain to roll off but the slope won’t get to be -6dB/oct because the zero is too close. It’s only one octave below the pole because the tube only has 6dB less gain with Ck removed. At lower frequencies still the zero cancels out the pole and the response is flat with the gain 6dB lower than the mid band value.
Dave
From: kg
Date: 1/28/2002 9:13 PM
Subject: Re: Poles and Zeros and Cathode Bypass Caps
It’s only one octave below the pole because the tube only has 6dB less gain with Ck removed.
hi dave,
i can give you multiple examples where the zero is more than 1 octave away from the pole. i know they’re not your quintessential 100k Rl, 1k5 Rk, 25uF Ck fender values, but they DO exist.
try crunching a 12ax7 with 50k Rl, 50k Rk, and a 5uF bypass cap. you’ll find the pole is about 60 (!) octaves higher than the zero. now, this example is NOT something you’ll typically find in your average guitar amp, but seriously, i have something very close to it in use in my preamp (100k Rl, 56k Rk, and 220nF Ck iirc), so it IS a viable circuit. all’s fair in rock and roll!
At lower frequencies still the zero cancels out the pole and the response is flat with the gain 6dB lower than the mid band value.
yes, the response will be flat below the zero (in the above-mentioned example the zero frequency will be 1Hz), but the midband gain will be far higher than this. with = value Rk and Rl we can expect the gain at dc to be -1x. we can also expect the midband gain to be at least -60x, even accounting for the loss of mu due to loading. that leaves us with a 60x gain differential, or about 35dBvg greater output at midband as a result of the Ck.
the increased output starts at 1Hz and will continue rising up the max the tube can provide, which will occur about 6 octaves higher, 6Hz or so. the reason it does not occur the full 60 octaves higher is that the tube has insufficient mu to continue rising at 6dB/oct… it craps out at 35dB.
ken
From: Michael Tousek
Date: 1/29/2002 3:54 AM
Subject: Re: Poles and Zeros and Cathode Bypass Caps
Thanks for that info, Ken.
Let me ask you this: for a typical 1.5k/100k 12ax7 gain stage with, say, a 1uF bypass cap, what will the frequency response look like?
I picture a relatively flat line coming from the lower frequencies, and then at some point (maybe 500 Hz or so?) it turns upwards and climbs for a while. Then, (maybe an octave or two later?) it levels out again and continues, in a relatively flat way, up into the frequencies that are beyond the speaker’s range.
I guess that’s a description of a shelving response. How would what you’re describing look?
Also, I would have sworn it was settled doctrine that a bypass cap boosts the effected frequencies by 6dB in a typical guitar amp gain stage. Did I just get hold of some bad info, or is 6dB about right as long as you’re talking about a garden-variety 1.5k/100k stage (or something similar)?
Thanks!
MT
From: kg
Date: 1/29/2002 12:59 PM
Subject: Re: Poles and Zeros and Cathode Bypass Caps
michael,
i don’t have the time right now, but you can crunch the numbers yourself…
the gain below the zero will be
gain(dc) = Rl/Rk.
the frequency of the zero is defined by the Rk and Ck, and nothing else. use the capacitive reactance formula of
Fz = 1 / 2 pi Rk Ck
here, which gives you Fz.
the frequency of the pole is defined by Rk and Ck and also Rk(int), which is the impedance seen looking into the cathode. this can be approximated by
Rk(int) = 1/gm.
to be a bit more exact, you should even take into account the load resistance (since the 1/gm approximation assumes no load in the plate circuit) and use
Rk(int) = (Rl + Ra) / (mu + 1)
to find the impedance looking into the cathode.
this impedance looking into the cathode is in parallel with the Rk, so find the resultant equivalent resistance using:
1/Rk(int) + 1/Rk = 1/R(tot)
then use THAT value for R in the capacitive reactance formula
Fp = 1 / 2 pi R(tot) Ck.
this is the frequency of the pole, Fp.
the number of octaves between the two is simple:
Fp/Fz.
you may find that there are combinations that offer many octaves of spread between the zero and pole. as i stated earlier, you reach a natural limit of voltage gain, defined by the tube and associated circuitry, which cannot be exceeded. this will be approximately
gain(ac) = (mu * Ra)/(Ra + ra).
you can compare this value for gain with the value for DC conditions (Rl/Rk) and solve for decibels of voltage gain (dBvg) with
dBvg = 20 log [(gain(ac)/gain(dc)]
the point at which this maximum gain is reached will be a certain number of octaves above the zero. this can be found with
#of octaves = dBvg/6dB per oct.
this is the “true” point at which the response begins to flatten, not due to the RC constants but rather the limitation of the gain stage.
i’d crunch some numbers for you but i don’t have the time right now. give it a shot and see what you come up with for various iterations.
ken
