The Valve Wizard

Pentodes have very high internal resistance and can be considered to be constant current devices. This makes them excellent for use as power valves. However, small signal pentodes can also be used in pre-amp stages, and they are favoured for their high sensitivity and high gain. If the load resistance is fairly large (and it usually will be) then they also produce a reasonable amount of odd-order harmonics on large signals, lending 'bite' and 'edge' to the tone.
The usual place for a small signal pentode is on the input of the pre-amp where the incoming signal is small, so sensitivity is a benefit. Drawbacks to using small signal pentodes are that they are usually more expensive and less readily available than triodes, and are more prone to being noisy and microphonic. The noise is due to 'partition' noise caused by the electron stream dividing to go to the anode and screen-grid. Partition noise is inversely proportional to frequency so reducing the bass response of the pentode (by partially bypassing the cathode say) will reduce noise. Small signal pentodes tend to be microphonic because they contain many, relatively large physical parts capable of vibrating, and because they are often operated at high gain (possibly close to a loudspeaker too!). Easy ways to reduce audible microphonics is to not use small signal pentodes in a combo amp, to mount the valve socket on a rubber grommet or similar, and to operate the stage at sensible gain levels (not more than 100 say). Additional gain can be provided by a preceeding, low noise triode stage, which has the bonus of allowing us to overdrive the pentode itself more heavily!

The cathode, control-grid and anode in a pentode serve the same purpose as in a triode, and the connections to these are the same as for a triode. The screen-grid and suppressor-grid are auxiliary electrodes and do not usually play a part in the AC part of the stage.

Supressor-Grid (g3): The suppressor-grid exists to repel "secondary electrons", which are deflected off the anode as it is bombarded, back to the anode again. Therefore, the voltage of the suppressor-grid must be negative with respect to anode to repel these electrons back to the anode and not toward anything else. For normal operation, the supressor grid should be connected to the cathode (sometime it is more convenient from the point of view of layout to connect it to ground). In many pentodes, the supressor grid is already connected to the cathode internally, so check the data sheet.

Screen-Grid (g2): The screen-grid exists to accelerate electrons toward the anode, and "screen" the control-grid against Miller capacitance. It is this extra acceleration that gives pentodes their high amplification factors. The screen voltage (Vg2) is usually made less than the anode voltage, either by a voltage divider from the HT, or simply by connecting it to the HT through a fairly large resistor (Rg2). Since some electrons are collected by the screen-grid, a small current flows through it, causing a voltage drop in the screen-grid resistor, placing the screen-grid at a lower voltage than the HT: Vdropped = Rg2 * Ig2. The screen voltage has much more control over how the pentode amplifies than the anode voltage. Lowering the screen voltage reduces headroom and compresses the grid curves. To see this happening, examine some anode characteristics at different screen voltages, HERE.
Do not exceed the rated maximum screen voltage or average power dissipation!

For normal operation the quiescent screen voltage will be less than the anode voltage. If the screen grid voltage is made too high then it will be at greater risk of over dissipation, especially during overdrive conditions. Although it is possible to design stages where the screen voltage is even higher than the anode voltage, it is neither recommended nor much use for guitar purposes.

Part of the difficulty in designing a pentode stage is that the grid curves change depending on the screen voltage. The data sheet may provide a few example graphs, but if they do not apply to the screen voltage you are going to be using then it will be necessary to draw the appropriate graph, by making an educated guess as to where the grid curves will fall (or use a computer model). Unfortunatly, we need to know the screen voltage to draw the graph, and we need the graph to find the screen voltage! Without a computer model, designing the stage becomes a trile-and-error process, but the following guide should help.

This example uses what is probably the most popular small signal pentode in audio amplifiers; the EF86 (6267). This pentode (and many other small-signal types) contains an internal shield which should be connected to the cathode or ground (this is shown in the diagrams below). There are also hundreds of different pentode types still available as new or used, including those not originally intended for audio applications, which can be experimented with.
Note that the EF86 data sheet quotes recommended component values that give very high voltage gain (around 200). These were primarily intended for sensitive circuits such as hifi phono-stages, not guitar amps. Such circuits were copied most famously by Vox, which have a reputation for being noisy, unless you specially select a very well-performing EF86. It is therefore strongly advised that you design your pentode stages for lower gain (not more than 100 say), more appropriate to guitar signal:noise levels, as this will considerably reduce microphonics and frustration in the long term!

In this example the pentode is to be at the input of the amp where HT is low, at 240V.

With a pentode, we can typically expect the quiescent anode voltage to be around a third to a half HT (slightly less than for a triode). If we assume it will be a half HT (120V) we can immediatly draw a graph showing the grid curves at that screen voltage, which will at least put us in the right ball park of operation:

We can now have a go at choosing a load line. For maximum output swing with reasonable linearity, the load line should pass through the 'knee' of the curves. If the load line passes above the knee then linearity improves, but signal swing is reduced; operation becomes more like that of a triode. If the load line passes below the knee then linearity gets worse, which is bad for hifi, but useful for guitar!
In this case we might try a load of 33k, but at 240V HT the load line is quite steep and would operate the valve rather close to its maximum rated current and power dissipation. A larger resistance of about 56k would be more suitable.

Suppose we want the load line to pass right through the knee, we can see that the 0V grid curve will need to cross the load line at about the 3.6mA mark (green dot). Looking at the mutual characteristics graph shows that this corresponds to a screen voltage of about 70V, so we can now re-draw the grid curves for this:

The graph now looks pretty good. Lowering the screen voltage has compressed the grid curves to suit the load line, so we can now choose our bias point. A bias of -1V looks reasonable, giving a quiescent anode voltage of about 130V, and anode current of about 2.0mA. However, before finding the value of the cathode resistor we must actually set the screen-grid voltage.

Setting screen voltage: The usual method is to add a screen resistor that will drop the necessary voltage to the screen-grid. To find its value we need only know the screen current that will flow through it, and apply Ohm's law.
As a rule of thumb, screen current is a fixed ratio of the anode current. The data sheet tells us that for an anode current of 3.0mA, screen current is 0.6mA. This is a ratio of: 3.0 / 0.6 = 5.
Therefore, if we wanted the screen to be at the same potential as the anode, we would use a screen-grid resistor five times larger than the anode resistor, to achieve the same voltage drop (280k would be the nearest standard). We can also use this ratio of 5:1 to find our safer screen voltage of 70V, since we know our quiescent anode current is about 2.0mA:
Ig2 = 2.0 / 5 = 0.4mA. We need to drop 240 - 70 = 170 volts across the screen resitor, so apply Ohm's law:
170 / 0.0004 = 425k
The nearest standard value of 470k will do. (330k would probably also do, resulting in a screen voltage of around 90V if we still biased to a Vgk of -1V.)

Alternatively, one or more zener diodes could be used to clamp the screen-grid at a fixed voltage. This would technically negate the need for a bypass capacitor, although we would still fit one to reduce zener noise.

Screen Bypass Capacitor: The screen bypass capacitor serves a similar purpose as the cathode bypass capacitor: it holds the screen voltage steady, to prevent internal negative feedback reducing gain. However, in the same way as for the cathode, this decoupling capacitor can be made small in value to boost higher frequencies, or omitted completely to introduce some internal feedback; many traditional guitar amp designs do not include this capacitor at all. In general, the screen bypass capacitor has a greater effect on the gain of the pentode than the cathode bypass capacitor.
The screen bypass capacitor can be connected to ground or to the cathode unless global negative feedback is being applied to the cathode (unlikely in a guitar amp), in which case the screen bypass capacitor MUST be connected to the cathode and not to ground, to prevent postitive feedback taking place.

To choose the screen-bypass capacitor properly it is necessary to know the internal impedance of the screen-grid. This can be derived using the anode characteristic curves for the EF86 in triode mode, by plotting a tangent against the chosen grid curve at the same quiescent anode voltage we are working with, i.e., Vg = -1.25V, Va = 130V. This yields a value for ra(triode) of 20k. The internal impedance of the screen grid is:
rg2 = ((Ia + Ig2) / Ig2) * ra(triode)
rg2 = ((0.002 + 0.0004 / 0.0004) * 20000
= 120k

This value appears in parallel with Rg2, making 88k, and this value is used to calculate the value of the decoupling capacitor. For a roll-off of 10Hz:
Cg2 = 1 / (2 * pi * f * R)
Cg2 = 1 / (2 * pi * 10 * 88000)
= 180nF
A more common standard value is 220nF, which would also do.
This is a rather tiresome process. The more reckless builder may prefer simply to use the value of screen resistor value to work out the screen bypass capacitor, and acknowledge that the roll-off frequency used in the calculation will turn out to be somewhat higher than predicted.

Biasing: The value of anode current read off the graph is not equal to the cathode current, which is of course the combination of the true anode current, plus screen current. To find the correct cathode resistor we must first find the current through the cathode alone; by adding the screen current (which we found earlier) to the anode current at the bias point, which we have already chosen to be -1.25V:
Ik = Ia + Ig2
Ik = 2.0 + 0.4
= 2.4mA.
The cathode resistor can now be found using Ohm's law:
1 / 0.0024 = 417 ohms.
So 330R or 470R should do. If particular accuracy is not required, the cathode resistor could be chosen using the anode current alone, ignoring screen current and still yield useful results. The pentode could also be biased using silicon diodes or an LED instead, and this would negate the need for a cathode bypass capacitor.

The data sheet tells us that if the anode is dissipating more than 0.2W, the maximum value of grid-leak is 3 Meg. We would normally just go with the typical value of 1Meg as this is an ideal input impedance in most circumstances.

Cathode Bypass Capacitor: The half-boost frequency due to the cathode bypass capacitor can be approximated as:
f = 1/(2 * pi* Rk * Ck)
In this case we will assume the design is for a high-gain lead guitar amplifier, where it is desirable to keep bass response minimal, and to reduce noise. For a roll-off of 300Hz we rearrange the above formula:
Ck = 1 / (2 * pi * f * Rk)
Ck = 1 / (2 * pi * 300 * 470)
= 1.1uF
The nearest standard is 1uF.

Gain: If both cathode and screen-grid are bypassed, the gain of the pentode is:
Av = gm * ra * Ra / (ra + Ra )
Since ra is very large, this can be approximated as:
Av = gm * Ra
Av = 0.0018 * 56000
= 101

If the cathode is not bypassed, the gain becomes:
Av(unbypassed) = gm *ra *Ra / [ ra + Ra + Rk (gm * ra + 1 )]
These formulae do not take into account the loading effect of the following stage, in which case Ra should be substituted for the value of Ra in parallel with Rl. They are also very innacurate unless very precise values for ra and gm are found. A better indication of gain can be gleaned simply by looking at the load line, which in this case suggests a gain of about 64, demonstrating the unreliability of our values for ra and gm.

Output Impedance: The anode output impedance is equal to the anode load in parallel with the pentode's internal impedance. Since its internal impedance is so large, it is usually assumed to be infinite, making the anode output impedance approximately:
Zout = Ra