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Back in 1871, long before the thermionic valve was invented, Maxwell [Max71] published the first study of the effect of a grid of wires on the field between two electrodes. Making some simplifying assumptions, he showed that the electric field as seen at the cathode is equivalent to a plate voltage of:
| $V_{eff}=V_{g}+\dfrac{V_{p}}{\mu}$ | where: | $V_{eff}$ = effective voltage seen at cathode $V_{g}$ = grid voltage $V_{p}$ = plate voltage $\mu$ = amlpification factor |
µ is a constant for a given electrode geometry. In other words, the actual plate voltage is divided by µ to get the effective voltage. For example, in a typical medium-µ triode under normal operating conditions, the effective voltage as seen at the cathode is only around 5V, even though the plate is at 100V or more.
Maxwell showed that µ can be calculated as follows. (More and more elaborate formulae for µ were developed throughout the life of the vacuum tube, and Maxwell's is not terribly accurate for real-life tubes).
| $\mu =\dfrac{-2\pi d_{gp}}{a \ln\left( 2 \sin \frac {\pi r_{g}}{a} \right))}$ | where: | $d_{gp}$ = distance from grid to plate $a$ = distance between grid wires $r_{g}$ = radius of grid wires |
In non-mathematical terms, this means that µ increases directly with the distance from the grid to the plate, and with the ratio of grid wire size to separation, $\frac{r_{g}}{a}$ (also called the grid pitch or shielding ratio). It also means that the value of µ is independent of the distance from the cathode to the grid. The variation of µ with cathode-plate distance is very visible. If you hold up high- and low-µ version of essentially the same tube, e.g. 6SL7 and 6SN7, or 12AX7 and 12AU7, you will see that the plate structure is much fatter in the high-µ tubes.
The assumptions that Maxwell made for his calculations were as follows:
- the electrodes are infinite, so that the behavior at the edges can be ignored
- the grid wires are small compared to their separation, i.e. the grid pitch is no more than about $\frac{1}{10}$
- the distance from the grid to the cathode is at least equal to the grid pitch
When the last condition is true, the effect of the grid wires is seen only collectively at the cathode, with no effect from individual wires even directly under them.
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| Figure 7: Triode equipotentials |
Figure 7 shows the field, by lines of constant potential, under varying grid potentials. It can clearly be seen that when the tube is conducting (i.e.the potential at the cathode is above zero) the field is uniform at the cathode. The effect of the individual wires falls off exponentially with the distance. At half-cutoff, the "bulge" due to the plate's field penetrating the grid is significant up to well over half-way to the cathode. This becomes significant for more modern tubes where the cathode-grid spacing is typically 60% of the grid pitch. Figure 8 shows the field in a cross-section of the tube, at a grid wire and midway between two grid wires, when no current is flowing, i.e. when there is no space charge. It shows that once past the grid, an electron is subject to the full potential gradient due to the plate voltage, but downstream of the grid this is essentially masked by the grid voltage, as predicted by Maxwell.
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| Figure 8: Section of triode electric field |
Current Flow: the Equivalent Diode
Using the effective plate voltage, we can start to calculate the current that will flow in the tube, following Childs Law, but we also need to know the value to use for the cathode-plate distance. A good approximation is given by:
| $d_{eq}=d_{cg}+\dfrac{d_{cg}+d_{gp}}{\mu}$ | where: | $d_{eq}$ = equivalent plate distance for diode equation $d_{cg}$ = distance cathode to grid $d_{gp}$ = distance from grid to plate |
(Spangenburg [Spang48] gives a more accurate, and more complicated, formula, but the result is only slightly different). Inserting this into Childs Law gives the complete equation for cathode current under given conditions:
| $I_{p}=P\left( V_{g}+\frac{V_{p}}{\mu} \right)^{\frac{3}{2}}$ | where | $P=\dfrac{2.335\cdot 10^{-6}A}{\left( d_{cg}+\frac{d_{cg}+d_{gp}}{\mu}\right)^{2}}$ |
$P$ is called the perveance of the tube, and is constant for any given tube geometry. A high-perveance tube is therefore simply one that will carry a high current. It can be seen that there are two ways to increase the perveance, either by increasing the electrode area or by decreasing the electrode spacing. The latter is more effective, but is limited by the mechanical construction techniques and achievable tolerances. Once the closest possible spacing has been reached, the only way left is to increase the area. This is why power tubes are physically large, ultimately leading to tubes like the monster WE212A which stands 13" tall.
Constants (so-called)
A triode is described essentially by three well-known so-called constants:
- Voltage amplification ($\mu$): as described above, the factor by which the grid reduces the effect of the plate voltage
- Mutual conductance ($G_{m}$): expressed in milliamps/volt (or nowadays milliSiemens, which means the same thing), the increase in plate current for a change in grid voltage
- Plate resistance ($r_{p}$): the effect output resistance of the tube. For small signals, the tube is equivalent to a voltage source in series with a resistance of this value
$$r_{p}=\frac{\mu}{G_{m}}$$
These constants appear in even the briefest data for a tube. Unfortunately, they are not at all constant. Figure 9 shows (for the 6SN7, a particularly linear triode) how they vary with plate current.
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| Figure 9: Changes of characteristics "constants" with plate current |
It can be seen that both $G_{m}$ and $r_{p}$ vary a great deal. In fact, this follows from Childs Law. With a little calculus (differentiating the formula with respect to $V_{g}$) and some algebra, we arrive at the formula:
$$G_{m}=\tfrac{3}{2}P^\tfrac{2}{3}A^\tfrac{1}{3}$$
In other words, the value of $G_{m}$ increases with the cube root of the plate current, and hence the value of $r_{p}$ decreases with the cube root of the plate current. Since $G_{m}$ is a key figure of merit for a tube, the manufacturer would always want to show the highest value, whch is to say at the highest rated operating current. More typical and reasonable operating levels reduce $G_{m}$ and increase $r_{p}$. For the 6SN7 shown, $G_{m}$ is 3.2mA/V at 16mA, but drops to 1mA/V when operating at 1.5mA. As in this case, $G_{m}$ typically drops faster than the formula predicts, particularly at low currents and high negative grid voltages.
The curves also show that $\mu$ is not really constant, either. As plate current drops from 5 mA to 1 mA, $\mu$ drops from 20 to 15, i.e. by about 25% - and the 6SN7 is a particularly linear tube in this regard. Newer miniature tubes show a steady drop over the whole operating range.
The explanation for this is not obvious, and the classic tube texts offer no explanation. Mainly it has to do with the changing shape of the electric field in the region between the cathode and the grid as the grid becomes more negative and the current drops. Especially with close electrode spacing (discussed in greater detail below), close to cut off, parts of the cathode below the grid wires are cut off while parts between the grid wires are still conducting - as can be imagined from Figure 7c. The parts which are still conducting have a lower value of $\mu$, which is why they have not yet cut off. (This phenomenon is called inselbildung, German for "island effect").
Another reason, particularly important at low currents, is that the value of $\mu$ is not constant throughout the tube. Variations due to the edges of the electrode structure, grid support wires, and irregular electrode shape mean that there are places where it is higher. As current falls, these areas are the first to cut off, leaving only the lower $\mu$ regions conducting. In consequence, the average value of $\mu$ falls with the reducing current, explaining the distinct tailing off at low currents.
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