Frem39] describes the theory which applies when the grid is closer to the cathode than its pitch.
Figure 10 shows how this applies to a 6SN7, which being an older design does not have especially close spacing. At -5.3V grid potential, the whole cathode is contributing to the plate current. As the grid becomes more negative, "inselbildung" starts - the parts of the cathode directly under the grid wires are in a negative field. By -10V less than half the cathode is still contributing. At -11.4V, the whole cathode is seeing a negative potential, and the tube is truly cut off, i.e. there is no plate current. It can be shown that the effect of inselbildung is to replace the 3/2 power in Child’s Law by a 5/2 power. Thus at high currents and small grid voltage, the tube obeys a 3/2 power, but as the grid is made more negative and current reduces, this gradually turns into a 5/2 law. This is the principal reason for the “tuck under” that plate curves show at large negative grid voltages.
From a practical point of view, at least as far as audio is concerned, the moral of all this is to operate tubes at as high a standing current and as low a bias level (i.e. closer to zero) as is possible for the circuit to operate correctly, so as to keep operation well into the 3/2 power part of the plate curves and hence reduce distortion, paticularly higher-harmonic distortion.
Filamentary TubesMost of the physics behind filamentary tubes is the same as for indirectly-heated tubes, but there are some differences. First, there is the question of the effective area of the cathode. An accepted formula for this is to use the length of the filament times twice the filament-grid distance [Spang48, p189].
The most significant difference arises because the voltage along the filament is not constant, but varies from one end to the other by the applied filament potential. Although this potential is small, it must be remembered that the effective plate voltage as seen at the cathode (filament) is also small. For example, a 300B operating under quiescent conditions of 350V and 90mA, with –60V on the grid, has a potential as seen at the cathode of around 15V, against a filament voltage of 5V. At the negative extreme of grid voltage, modulated by the signal, this effective voltage will drop close to or even below 5V.
When the effective plate voltage is less than the filament voltage, only part of the filament contributes to the plate current, i.e. the part which is still more negative than the filament. Furthermore, the current varies along the filament. The effect of this is that the current becomes dependent on the 5/2 power of the effective plate voltage, rather than the 3/2 power. As the plate voltage increases beyond the filament voltage, there is a gradual transition between the 5/2 power and the 3/2 power, which is approximately given by the formula [Dow37]:
Using AC rather than DC for the filaments does not reduce this effect. At any given instant, even with AC heating, there is a potential gradient along the filament (except of course for the moment when the filament voltage passes through zero). Some of the time it is greater than the equivalent DC voltage, and some of the time it is less, but taken through the whole AC cycle the net effect is essentially the same.
Contact PotentialAny two different metals placed in contact with each other generate a potential difference. This is the underlying principle of all batteries, as well as the thermocouple. This difference is due to the different energy levels of the electrons in the two metals, and is called the contact potential of the two metals. The reasons are similar to the work function which determines electron emission, although the two are not the same. In fact, this effect applies even if the two metals are not in contact with each other, applying in this case to the electric field between them. Thus the grid and cathode of a tube, typically made of nickel and barium/strontium oxide respectively, have a small contact potential, which serves to change the effective grid potential. This contact potential is typically less than 0.5V.
Transit TimeThe electrons must take a finite amount of time to travel from the cathode to the plate. This time is referred to as transit time, and is sometimes invoked to explain various phenomena relating to audio. Transit time did indeed become of practical importance when tubes were first used to build VHF and UHF equipment, and it ultimately sets a limit to the frequencies at which they can usefully operate (in the region 1-2GHz). The transit time can be calculated to be around 1nS from cathode to plate, which at audio frequencies is clearly not relevant.
The TetrodeThe triode was the first amplifying device to be built, but at radio frequencies it suffers from a grave disadvantage because of the Miller Effect, which gives it a large effective input capacitance in the conventional common-cathode circuit. To avoid this, the tetrode was invented, having a second grid (the screen grid) between the triode grid (called the control grid in multi-grid tubes) and the plate. This grid is connected to a positive voltage close to the plate potential, but grounded to high frequencies through a decoupling capacitor. This results in an electrostatic shield which reduces the effective control grid-plate capacitance to a very low value.
A secondary effect of the screen grid is to reduce dramatically the influence of the plate voltage on the current flow, since the cathode is shielded from the plate by not one but two grids, and their screening effect is multiplied. As a result, the plate curves of a tetrode are very flat, as seen on the right-hand side of Figure 11. This corresponds to a very high value of plate resistance, as compared to a triode.
Some of the current from the cathode goes to the screen grid rather than the plate. The proportion depends on the shielding factor of the screen grid and on the relative potentials of the two electrodes, in much the same way as for a triode operated with a positive grid, and is typically 10-25%.
The PentodeThe solution to the tetrode’s problem was to introduce a third grid between the screen grid and the plate. Called the suppressor grid, this is always connected to the cathode and hence appears negative both to the plate and to the screen grid. By this means, secondary electrons emitted from both of these electrodes see a field which sends them back where they came from, regardless of the relative potential between the electrodes. Thus the problems of secondary emission are eliminated. The suppressor grid has no effect on the flow of current, since by the time electrons reach it they have been sufficiently accelerated by the screen grid that they simply pass between the grid wires. They are slowed down but not stopped by the grounded suppressor grid.
Since the cathode is shielded from the plate by no less than three grids, the effect of plate voltage on current flow is negligible in the pentode, resulting in plate curves that are even flatter than for the tetrode.
The Beam TetrodeThe beam tetrode exploits an alternative way of avoiding secondary emission problems, without the manufacturing complexity of using a third grid. It was observed in the 1930s that if the distance from the screen grid to the plate is large enough, the space charge of the electrons flowing in this region can depress the potential substantially without having an actual electrode. This is the basis of the beam tetrode. The reduced potential in this region serves the same function as the suppressor grid, causing secondary electrons to turn back to their origin and avoiding their effect on the electrode currents.
To make this effect work in practice, three things are necessary. First, the electron flow must be confined to a narrow beam, otherwise the space charge spreads out parallel to the plate and the effect is lost. This is achieved by the beam electrodes, carefully shaped plates connected to the cathode and placed either side of the electron path between the screen grid and the plate. Second, the electrons must flow in clean “sheets”, which requires that the grid wires of the control grid and the screen grid be in line. This is not surprisingly a tricky manufacturing problem. Finally, the electrode dimensions and spacing must be carefully calculated.
Although there are slight differences in the detailed operation of pentodes and beam tetrodes, for most practical purposes they can be considered as the same. Indeed, there are tube types which some manufacturers built one way, and others in the other way. Eric Barbour [Barb97] mentions that the EL84/6BQ5 was made in both ways.
Spang48] and Beck [Beck53]. Unfortunately these are extremely difficult to get hold of. These can really be regarded respectively as the US and UK “bible” on the subject. Spangenburg does however have a number of baffling minor errors in the transcription of formulae from other sources, which means that reference to original sources is required for certainty.
Dow [Dow37] seems to be easier to find, and gives many of the basic principles as well as a good introduction to the use of tubes in circuits. There is also a later book by Spangenburg [Spang57], although shorter than the first and covering semiconductors as well as tubes, which is quite adequate and seems easier to find. Reich [Reich41] is fairly basic but has the advantage of being available in a reprint [Reich95]. Valley & Wallman [Valley46] deals largely with DC and pulse amplifiers using tubes, and covers several topics such as low-level amplifiers in more detail than elsewhere. Mitchell [Mitch93] gives the most comprehensive tube data available, unfortunately only for a small selection of tube types. Smullin [Smullin59] gives a comprehensive treatment of all aspects of noise in vacuum tubes.