Instead of my posts being buried in cyberspace to never be found again, except by my laborious and repeated typing... ya know I wouldn't mind going on to beat in the point of newbies coming in not knowing a damn and wanting it all told to their face, but... I understand this fact of life, so for posterity I shall list some of my longer and interestinger posts here. Enjoy.

## Tube Amplifier Bandwidth

In response to...
What would be the rise time of an output tube? Isn't this a function of the load it is operating into and the time constant created by this circuit? Is rise time even a relevant concept of a tube on its own, independent of the circuit it is in?

Ah, here's something I know quite a bit about.

First of all, realize that this is an utterly useless topic of discussion in regards to audio amplifiers. I joke with a friend of mine that we like to design fast audio amplifiers because it lets us brag about the size of our genitalia...

A possible exception is class D, but here on the tube forum, we don't talk about that much (well, people other than myself don't!), and that's a different sort of signal anyway (the carrier's rise/fall time doesn't make it to the output, or at least it's not supposed to; it is a real signal on the plates and grids, however, and thus worthy of consideration).

A very effective model of a tube takes into account all capacitances (including miller C!) and resistances (i.e., plate resistance in parallel with load resistance), treating the tube as, say, a CCS in parallel with this impedance. (A CCS is fair because the plate impedance is already taken into account (triode or otherwise), and because amplifiers are most often modeled as CCSs for physical as well as theoretical reasons.) So you've got the grid swinging some amount of voltage, and this causes so much change in plate current (delta Ip = delta Vg * Gm), which therefore causes the capacitance to start to change its voltage (I = C * dV/dt), and the resistor to [eventually] come to a different equilibrium voltage (V = I*R). This is why you always see those nice smooth RC exponential decays on tube-amplified square waves. By the way, this model is linear, and tubes are reasonably linear (they don't hard-saturate the same way transistors do, nor does capacitance change with voltage), so you'd expect to see the RC time constant remain constant for all signal levels -- and this is fairly well observed, rise time is actually the characteristic rise time, not a slew rate per se. So it's not a bad model at all, very simple and very useful.

A more thorough model might include transit time effects and lead inductance, but these generally aren't important below 30MHz. In sub-VHF use, tubes can be considered as instantaneous transconductance amps driving the RC load I just described.

Therefore, switching time is effectively proportional to R*C. In general use (like many of the circuits you see on this forum), bandwidth typically ends at 50-200kHz (higher where lower resistance loads and lower capacitance tubes are used), with intentionally high bandwidth designs going well beyond 1MHz (Tektronix made some excellent amps -- using regular old 12AT7s and 6AU6s -- that did up to 20MHz. Notice they cheat to obtain the last octave and a half by using peaking inductors!). I don't know that resistor-coupled tube stages go (or can go!) beyond 100MHz. Certainly, that would require specialty tubes (which actually sounds like something Tek might've made, if higher performance transistors hadn't come along in the 60s to replace them). Oh, and notice RF amps cancel the capacitance -- at one frequency! -- using inductance. This allows rather pedestrian tubes, like the 12AU7, to perform well in osc/amp service over 100MHz (at reduced ratings). This is not possible in wideband use, which is what we're doing.

Let's take some values on the 300B:
Cout: 4.1pF (Cpk)
Cgp: 15pF
Cin: 8.5pF (Cgk)

At a typical gain of 3.0 let's say (a bit less than mu, typical for a loaded stage), Cgp is effectively 60pF, so Cin(tot) = 68.5pF and Cout(tot) = 64.1pF. Let's say you have an even 3kohm load resistor for the plate load, the tube is biased around 300Vp, the supply is 500V and plate current is around 67mA. Assume the resistor has negligible capacitance and your scope probe adds 10pF (reasonable guess for a 10x probe?). The output impedance is thus 740Ω || 3000Ω || 74pF, or 593Ω || 74pF. So, R*C = 44ns. Risetime is often defined as the time taken for voltage to swing from 10% to 90%, but one time constant is only 63%; two time constants is pretty close (85% vs. 80%), so the expected risetime will be around 100ns. The -3dB rolloff is 1/(2*pi*R*C) or 3.6MHz. Not bad, not bad at all.

However, this is assuming your grid is driven from a perfect voltage source. It takes a lot of current to make the grid change voltage instantly! You have pretty good theoretical plate bandwidth, but you can't actually get all of it.

Let's say your drive comes from something like a 12AU7 cathode follower (12AU7? Sure, why not- in class D, the driver tubes don't matter, they're *supposed* to distort!). Then you'll get ballpark maybe 10mA drive in both directions, which can slew the grid voltage at about 140V/us, enough to turn it fully on or off in about a microsecond (delta V = 140V or so; this would go a lot faster if the output tube had more gain). (A better driver choice might be 6DJ8 or 5687, which have higher gain and current capacity. You still need a fast gain stage to generate the swing, too; a cascoded pentode would be delicious.) As a result, you might only get 500kHz bandwidth, if that. This is pretty close to real observations. If the driver has even less current available (6SN7 plate output is very typical), and your output load has more capacitance (an OPT might cost you a whole nanofarad!), your real bandwidth might drop to a paltry 30kHz, which is, in fact, exactly the observation in many cases. Fortunately, this is still good enough for audio, so we aren't at all worried -- except when discussing the size of one's genitalia.

Tim