Tubes 201 - How Vacuum Tubes Really Work, Part 6: Initial Electron Velocity
The derivation of Childs Law assumes that electrons are emitted from the cathode with zero velocity, but in the description of emission we have seen that this is not the case. As shown in Figure 4, emitted electrons have a distribution of energy which falls off rapidly. This means that all of them have enough energy to move away from the cathode, creating a dense cloud of electrons just ahead of the cathode. This in turn creates a dense region of space charge, and it is this space charge which, by repelling the emitted electrons, reverses their direction and sends them back to the cathode. Only the small minority that are emitted with enough energy to penetrate this strongly-charged region are able to reach the attracting field of the plate and hence form part of the plate current.
This affects the current flow in the tube in a number of significant ways. The blue line in Figure 6 shows the electric potential in the space between the cathode and the plate (of a diode), taking into account the initial velocity. Initially this potential drops to a negative value, which then starts to rise. The minimum of the negative value forms a virtual cathode, and it is from this virtual cathode that the calculation of current flow must be taken into account. (Both the potential of the virtual cathode and its distance from the physical cathode depend on the current flow, which creates an obvious problem of circularity for computing the current.) The effect is that, firstly, the effective cathode-plate distance is less than the physical distance, and secondly, that the effective plate potential is increased. The latter is significant, considering that the effective plate potential, as seen at the cathode, is only a few volts.
Even beyond the virtual cathode, some of the electrons still have some residual energy. The effect of this is to increase the current beyond what Childs Law predicts. The full physics of what is going on here is complex, and leads to some alarming mathematics, which were first fully worked out by Langmuir in 1923 [Lang23], and can also be found in [Dow37] and [Spang48]. Approximate, but somewhat frightening, formulae are shown below. The math-adverse should skip to the following paragraph immediately.
| $d_{vc}=\dfrac{0.0156}{\sqrt{1000I_{p}}}\left( \dfrac{T}{1000} \right)^\tfrac{3}{4}$ | where: | $d_{vc}$ = distance of virtual cathode from physical cathode (cm)
$I_{p}$ = plate current
$T$ = cathode temperature (°K), typically around 1050°K |
| |
| $V_{m}=\left( \dfrac{T}{5040} \right)\log\left( \dfrac{I_{0}}{I_{p}} \right)$ | where: | $V_{m}$ = effective virtual cathode potential |
| |
| $I_{p}=\dfrac{2.335\cdot 10^{-6}A(V_{p}-V{m})^{\tfrac{3}{2}}}{(d_{cp}-_{vc})^{2}}\left( 1+0.0247\sqrt{\dfrac{T}{V_{p}-V_{m}}} \right)
$ |
It can be seen that evaluation of these formulae requires knowledge of the emitted current from the cathode ($I_{0}$). This is a hard quantity to measure, and furthermore it decreases continuously during the life of the tube. For a new tube a typical value is around 1A/cm2, i.e. around 100 times the plate current. The most important equation above is the bottom one, which gives the plate current in terms of potentials. It is rather frightening, so start with the left part. This simply Child's law, as seen earlier, with a correction for the potential and distance of the virtual cathode. The final term is the correction for initial velocity, which at low currents represents a substantial increase in plate current.
Substituting some typical values into these formulae shows that the reduction in effective cathode-plate separation due to initial electron velocity is around 0.1mm, while the increase in effective plate (or combined plate and grid) potential is around 0.5V. The increased current due to initial velocity is around 10-25%. These figures increase substantially as the tube gets close to cut-off, especially the distance to the virtual cathode.
These formulae explain why the heater voltage has an effect on the operation of a tube, which is not explained by Child's Law. As heater voltage is increased, two things happen. First, the emitted current is increased, thereby increasing the effective plate voltage. Second, the energy of the emitted electrons (due to the higher temperature) is increased. In fact, a 10% increase in heater voltage gives about a 5% increase in plate current.
Because the virtual cathode is negative with respect to the cathode, some current will flow even if the plate is at the same potential as the physical cathode. Mitchell [Mitch93] shows that a 12AU7 with 0V on both the grid and the plate passes a current of 90µA, while the 6DJ8 under the same circumstances passes 300µA, the latter due to the closer electrode spacing. (Anecdotally, the author had a Leak FM tuner which gave low and distorted output. Investigation showed that a broken wire meant that the first stage, an ECC85, was operating with 0V on the plate. But it was still passing a signal).
The formulae given above apply to a plane electrode structure (i.e. Figure 5b). If the electrodes are truly cylindrical (Figure 5a) then different, rather more complicated, formulae apply, but the general principle remains the same. However, there is one significant difference, which is that the distance from the true cathode to the virtual cathode ($d_{vc}$) is smaller with cylindrical tubes. This may be the explanation for the reported qualities of early triodes such as the 76.
If the plate is made negative, some current will still flow. As it decreases, the virtual cathode will move further away from the physical cathode, until eventually it reaches the plate. From this point on, a different set of rules apply and the current decreases exponentially with increasingly negative plate voltage:
| $$I_{p}=I_{0}\cdot e^{\left(\dfrac{11605V_{p}}{T}\right )}$$ |
(Note that in this formula Vp is negative). Thus in principle the current is never reduced completely to zero, but at Vp=-1.3V it is reduced to 1µA, and at Vp=-1.9V to 1nA.
The pool of electrons in region between the physical and virtual cathodes is sometimes stated to form a reserve that can supplement the cathodes emission during current peaks. This is a myth; these elec; these electrons have a very short lifetime before they return to the cathode, and correspond to less than 1nS of emission.
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