Name: Ken Gilbert
 E-Mail:
 Date/Time: 8/16/99 9:25 PM
 Subject: More (LOTS more) on cathode followers (Hey Steve!)


Ok Steve, here's my attempt at a more thourough description... It may ramble a bit,
but try to hold on 'till the end ;-/

A couple of things about triodes in general... typically the cathode will "rise" to a
voltage a few volts higher than the grid. This voltage difference could be thought of as
the "bias voltage" aka Vgk. 

This voltage difference between grid and cathode will occur only if there is some
series resistance for the cathode current flow through (since it develops this voltage
difference by pulling current through a resistance). If there's no resistor, the cathode is
obviously at ground and will not rise higher, no matter how much current it pulls.

So then, for a given value of grid voltage (like you would have in a DC'd CF), you can
easily estimate the current through the tube by using V=IR. For the "V" use the grid's
voltage +5 or so (for an AX7). The "R" will be the value of cathode resistor.
Rearrange as necessary for determination of other variables.

Let's do an example of designing a DC'd CF for maximum gain and output capability.
Assume a 12AX7 gain stage, with a B+ of 300Vdc.

Assume that Rk is 2K2, a common enough value. With an Ra of 220K, Ip is about
0.63 mA. This current causes a 138V drop across the Ra, leaving the anode at about
160V. Bypass the Rk with a 25 mic @ 25 V cap, and it's pretty much set up for max
gain and max clean signal swing output. [You COULD increase B+ at this point, until
the plate sat at 300VDC at idle. That would increase the signal swing output
significantly, but let's not do that.]

Well bypassed, this stage will have a voltage gain of about 37 dB or so (about 72x),
and a clean pk-pk output of 60V or so. The plate voltage is sitting at about 1/2 B+,
which is ideal for max clean signal swing.

So now we take the anode, and connect it DIRECTLY to the grid of a following tube.
We also take that following tube and connect its anode right to B+, or 300V. There is
some, as of yet, unknown resistance in between the cathode and ground.

What will happen here? Well, some people think of the CF as biased between the grid
and PLATE, as opposed to the cathode. This is true to some extent in that as the
positive excursion of the grid approaches the plate voltage (which is unchanging) the
tube will run out of steam, and will not be able to draw its cathode and plate voltages
together. At that moment, the tube is operating as if it had a very low effective plate
voltage, which is, after all, the voltage from anode to cathode. The output at this point
(or even NEARING this point) will be severely distorted.

So what about the cathode itself? It has an output impedance of approximately 1/gm.
Assuming a 12AX7, that has a transconductance of about 1650 micromhos at 1.2 mA
of plate current. Inverting that (0.001650 mhos) gets you an output Z of approximately
600 ohms. Pretty neat, hunh? That's a very low output impedance, able to drive quite a
low impedance before it gets really nasty. MUCH lower than a plate loaded 12AX7!

Remember that gm will increase along with plate current. So if you DECREASE plate
current, you will DECREASE transconductance, INCREASING the output
impedance of the cathode follower. That's not usually what you want to do with CF's,
since they're most useful as drivers, and drivers are best with low output impedances.
So let's try to maximize gm.

Obviously, INCREASING the plate current is what we need to do. But there is a limit
here--well two really--one is the dissipation of the tube. The other is the max cathode
current, which is 8 mA as per the data sheets. That represents the total emissions
capability of the cathode itself... more than that and the lifespan will be decreased
greatly. It is irrespective of plate voltage. So we know we can't exceed 8 mA as a
definite max.

How about dissipation? We know that the grid will be sitting at about 150VDC [so I
rounded it off a bit!]. That means that the cathode sits a few volts higher than this, or
about 155VDC. [The actual difference in grid-cathode voltage depends on the
transconductance of the tube, which, as you know, depends on the plate current! So
we can't really know it for sure.]

That means the voltage from cathode to plate will be 145VDC. The maximum anode
dissipation of a 12AX7 is 1.0W. At that voltage, the plate current to create 1.0W of
dissipation is 1W/145V, which is 0.0069A, or 6.9 mA. This represents the other
maximum current which may be passed through the tube at this particular effective
plate voltage.

Naturally we must take the lower of the two maximums, which gives us 6.9 mA. Now,
it's never a great idea to run tubes at max dissipations, so we really have to derate that
a bit. Let's make it a max of 5 mA, a nice round number. At 145 plate volts, that gives
a dissipation of 145V * 0.005A, or 0.725 W at idle... reasonable enough to expect
normal life out of the tube.

OK, we need to pull 5 mA through this CF stage. How do we ensure that? Well, if the
cathode is sitting at 155VDC, and ground is 0VDC, and there's 5 mA flowing, then we
have good ol' Ohm's law to find a value for Rk. This is the same thing I talked about
at the beginning of this post, only then I was using a common cathode stage. In this
case, it's no different: 155V / .005A = 31K. This is our cathode resistor to allow 5 mA
to flow. Make it bigger, and the current goes down--smaller and the current goes up.

31K is large enough so that it appears much larger than the cathode impedance itself.
This allows the cathode to avoid being "loaded down" too much. Too low of a
cathode resistor (assuming you don't burn up the tube first) will cause the loss of
signal voltage associated with ALL cathode followers to INCREASE. Your gain will
become SMALLER; That's not good. Luckily for us, 31K is large compared to 600
ohms, so the loss through the stage will not be too great.

We have optimized the value of cathode resistance to allow a decently large amount of
current to flow in the second, cathode follower connected stage. This has the effect of
minimizing output impedance, but it also has another (somewhat associated) benefit:
improving the slew rate performance of the CF output.

Slew rate is a fancy term to describe how fast the output voltage can change with
respect to time. A high slew rate is associated with high current stages, since a high
current stage is better able to sink and source the currents necessary to drive
capacitive loads. EVERY wire in your amp, and EVERY component, has some
associated capacitance which must be driven. Any time you attempt to change the
voltage of a circuit node which has a capacitance, current must first flow before the
voltage will change. This is the very definition of capacitance.

This would be more of an issue with a long interconnect cable, say an effects send, or
a line output, which has a rather large amount of capacitance to overcome in the cable
itself. The driver with a higher slew rate is better able to drive this capacitance,
because it has the lower impedance (and therefore the current) to "back it up." This
means that the natural, first order HF rolloff will be pushed UP in frequency, extending
your frequency response and your overall fidelity.

The CF can only sink as much current as that which is pulled up through it. This is
because of the passive pull-down nature of the resistive load. It can _source_ much
more than this value. [Remember that electrons flow opposite to conventional current
flow. Therefore sinking current means "pushing" electrons, and sourcing current
means "pulling" electrons from the load.] So essentially by maximizing the current
through the CF we have maximized it's ability to deliver current swings into the load. It
is simply a better driver now.

The increases in gm with plate current pretty much max out above 2 mA with the
12AX7. This can be easily seen on the graph of transfer characteristics, which is
nothing more than the graph of gm at certain plate voltages. See the Amperex data
sheets (somewhere out there on the web) for the graph.

That's all I have time for for now. I've probably brought up more questions than I've
answered, but that's normal enough!

KG