"Tubes 201" - How Vacuum Tubes Really Work
Copyright © 2003, John Harper
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Table of Contents
Introduction
Emission
Physical Construction
Space Charge and Current Flow
The Triode Field and Operation
Initial Electron Velocity
Noise
Other Things
Multi-Grid Tubes
Further Reading and References
Reassurance for the Math Averse
There are lots of equations in this article. Be assured, if you don't like math, you can skip them. The text tells you the important things that the equations mean. Of course, if you're happy with math, you'll get more from this article. (And there isn't any real math, i.e derivations of things).
1. Introduction - the Basics
Everyone knows how a tube works. Current passing through the filament heats it up so that it gives off electrons. These, being negatively charged, are attracted to the positive plate. A grid of wires between the filament (or cathode) and the plate is negative, which repels the electrons and hence controls the current to the plate. For many purposes that’s all you really need to know. This article looks beyond the basics, into what makes a tube behave the way it does.
As a start, let’s recap the well-known properties of tubes, which will probably be familiar to most people reading this article. We will focus on the use of tubes for audio and will look mainly at triodes, with a look later at multi-grid tubes for audio.
The behavior of a triode is fully described by its plate curves, as shown in Figure 1. These show the plate current as a function of plate voltage (on the horizontal axis) and the grid voltage, becoming more negative as we move to the right of the family of curves. The curves for audio tubes generally only show negative grid voltage and positive plate voltage, although it is common to operate transmitting tubes with positive grids, and it is even possible (though rarely useful) to operate with a negative plate. These curves are just a way to represent a three-dimensional surface on the printed page, and in the old days people would even build models from plaster to represent this (e.g. in [Chaff33]). Nowadays we can just ask the computer to draw it for us, as shown in Figure 2. Mathematically, this is a representation of a function which takes two arguments (grid voltage and plate voltage) and gives the plate current as its result. The standard texts (e.g. [Lang53] show how to use these curves to establish the operating conditions for a tube.
Figure 1: Plate curves for typical triode Figure 2: Surface representation of triode characteristics
There are other ways to represent the same function. For example, we can draw the “transfer characteristics”, which show plate current as a function of grid voltage, with different curves in the family for different plate voltages. It is also possible to draw curves which show derivatives, such as the values of plate resistance and mutual conductance under different conditions. However these are not different characteristics, just different ways of looking at the same data.
One thing that leaps out is that the plate curves are just that - curves. In fact an ideal amplifying device, at least for audio, would show parallel, equally-spaced straight lines rather than curves. Unfortunately no real-world amplifying device can do this, since they all depend on physical phenomena that result in more complex transfer functions. Incidentally the obsession with linearity is somewhat peculiar to the world of audio. In other branches of electronics, such as RF and video, linearity is not so dramatically important since reasonable non-linearity can be filtered out either electronically (e.g. by tuned circuits) or by the receiving device (e.g. the eye). Only the ear can detect such tiny traces of non-linearity.
The plate curves follow quite closely a 3/2 power law, in which the current increases as a function of the 3/2 power of either the grid or the plate voltage. This is especially true at high currents and in the lower (closer to zero) range of grid voltage. While this is not linear, it is closer to linearity than any solid-state amplifying device, which is the main reason why tube amplifiers are nowadays often considered (and certainly by most of you who will be reading this) to be sonically superior. At low currents, especially as the grid becomes more negative, the 3/2 power law no longer applies and the plate curves “tuck under” very noticeably. The curves are not equally-spaced, even at high currents, but rather get closer together as current increases. All of this is causes extra non-linearity and distortion. This is more true for some tubes than for others, and later on we will look at some of the reasons.
In the sections which follow we take a look at the reality of tube design, construction and operation, which will explain why tubes behave the way they do. On the way we will explode a few popular myths.
2. Emission
Underlying all tube operation is the fact that any metal is continuously emitting electrons. Both the number and the speed with which they are emitted increases very strongly with temperature, although in fact emission takes place at anything above absolute zero (-273°C). To understood emission, we have to look at what is going on inside the body of the metal. In any metal, there are one or two electrons that can easily be detached from an atom, so that inside the solid metal there is a kind of sea of electrons floating around independently of any particular atom. The latter are fixed in place inside the crystal structure and do not move about at all, although they vibrate in place. This sea of electrons is common to all metals, and indeed is really the defining characteristic of a metal and explains many of their familiar properties such as electrical conductivity and the fact that they are shiny.
Since the electrons are not attached to any particular atom, they move about constantly, very much like the molecules in a gas. The average speed of the electrons increases with temperature, but because they are constantly bouncing off of the atoms and each other they do not all have the same speed but rather obey a statistical distribution law.
If an electron happens to be going towards the surface of the metal, then it will naturally tend to fly right out through the surface. However there are powerful forces trying to stop it, for the simple reason that there are positively charged metal atoms inside (because they have lost one or two electrons to the “electron sea”) and none outside. Thus an electron approaching the surface is slowed down, and only those having enough energy can escape. The amount of energy required is called the “work function”, and varies for different metals.
This is a convenient point to say how electron energy is measured. First of all, the energy of an electron corresponds directly to its speed. This follows the same law for kinetic energy as anything else, such as a car:
where: E = energy
m = mass
V = velocity
In this case, m is the mass of an electron, which is about 10-30 kg. Energy is normally measured in Joules, but for electrons this is impracticably huge. Instead we use “electron Volts” (eV). One eV is the energy that an electron acquires when it is accelerated through a potential field of one Volt. It is about 10-19 Joules, and corresponds to a speed of about 800,000 meters/sec.
The work function of a metal is expressed in eV. For tungsten, it is about 4.5eV. Any electron having less energy than this will not manage to escape, but will be turned around by the electric field close to the surface and will return into the body of the metal.
The electrons escaping from the metal correspond to an electric current, and this current is given by Dushmann’s Equation:
where: I0 = emitted current
A = a constant, 120.4 A/cm2
T = temperature expressed in°K (i.e. °C + 273)
w = work function of emitting metal
e = 2.71828…
The striking thing about this equation is the exponential element, which means that emission increases very rapidly with temperature. Figure 3 shows the emission of a tungsten filament as a function of temperature. Even a small percentage change in temperature results in a big change in the emitted current. For an oxide-coated cathode under typical operating conditions, a 10% increase in temeprature increases emission by about a factor of 3.
Figure 3: Electron emission as a function of temperature
The electrons that do manage to escape have the same distribution of energy as they did inside the body of the metal. Some of them flop exhaustedly from the surface, while others still have considerable velocity. This becomes important when examining the behavior of the tube. The distribution of energy (and hence speed) obeys the equation:
where: p =proportion of all electrons having energy greater than V
qe = electron charge, 1.602·10-19 Coulomb
k = Boltzmann’s constant, 1.38·10-23 Watt-Sec
T = temperature expressed in °K (i.e. °C + 273)
e = 2.71828…
It is not a coincidence that the exponential element here closely resembles that in Dushmann’s equation. Figure 4 shows this distribution graphically. The great majority of electrons have low energy levels, and the average for an oxide cathode is only about 0.1eV, but there is no upper limit on the energy that a single electron may have. For example, about one electron in a billion is emitted with an energy greater than 1eV.
Figure 4: Electron velocity distribution: proportion of emitted electrons having given energy
Early tubes used solid tungsten filaments. Tungsten has a high work function, and there are other metals which are much more suitable in this respect (for example caesium, whose work function is only 1.6eV). However tungsten has the great advantage of a high melting point - other metals would melt long before they gave adequate emission. A tungsten filament has to be operated at about 2700°C, which is the same as a light bulb. The amount of heat thrown out by any hot object increases with the fourth power of its temperature, which means that a great deal of power (i.e. filament current) is required to replace this lost heat and remain at this temperature.
It was fairly soon discovered that the addition of a small amount of the obscure element thorium (about 1%) to tungsten greatly reduces its work function, to about 2.6eV, allowing filament operation at around 1900°C. This reduces the required power by about a factor of four. Such tubes were called “dull emitters”, because compared to a tungsten filament (or a light bulb) they were much less bright.
Later it was found that a surface coating of barium oxide (or a mixture of barium and strontium oxides) gave even better results. This is because the oxide is no longer a metal, and the energetic electrons within the body of the filament can escape through the oxide layer at much lower velocity. In fact, the oxide layer is a n-type semiconductor, i.e. one having an excess of electrons, and its behavior is due to this. Oxide filaments have a work function of about 1.1eV, and can be operated at around 700°C. It is this oxide coating which makes filaments and cathodes appear white. There are however disadvantages. The oxide coating is mechanically fragile, and can be damaged as a result of gas in the tube or by vibration or shock. This is why high voltage tubes (such as the 211 and 845) still use thoriated tungsten. It is also relatively volatile, and slowly evaporates from the surface of the filament from where it is deposited in undesirable places such as the grid wires. Emission from an oxide surface is much more complex, in physical terms, than from a pure metal, although it does still essentially follow Dushmann’s equation. Overall, though, the advantages of oxide-coated filaments and cathodes for small-signal tubes are overwhelming, and no other materials have been used since the 1930s.
Electrons within the body of the cathode are travelling in all directions, and in consequence so are the ones that are emitted. It turns out that the average lateral (sideways) velocity of the electrons is about the same as their average forward velocity. It is therefore wrong to think of them all travelling along the shortest straight line from the cathode towards the plate.
3. Physical Construction
There are many different ways that a filament (or cathode), a grid and a plate can be constructed and put together, and many of them have been tried over the years. From here on, we will concentrate mainly on indirectly heated tubes, having a distinct heater and cathode, rather than filamentary tubes. These appeared in the early 1930s, primarily as a way to allow the heater to operate from AC rather than DC. The only disadvantage of indirect heating is that it is less efficient, i.e. more energy input is required for operation at a given temperature. This is why tungsten and thoriated filament tubes (e.g. 211, 845, SV572) are still directly heated, due to their much higher operating temperature, and also why battery tubes were. For oxide cathodes, the best known filamentary tube in common use today is the 300B, together with a few other power types.
Figure 5: Types of triode construction
There are three common tube structures, shown in Figure 5. 5a shows the cylindrical structure, in which the cathode, the grid and the plate are made as three concentric cylinders. This structure has been little used since the 1930s, presumably because it is more expensive to manufacture. (An exception is rectifier diodes, which are almost invariably built like this). It does however have some theoretical properties that may account for the sonic reputation of the classic model 27 and 76 triodes, which are built this way. The grid is wound as a spiral, generally with two vertical support posts. The grid wire is rigid, and holds its shape once it has been formed into shape.
Figure 5b shows the flat structure used in power tubes such as the 6AS7 and (with a filament rather than a cathode) the 2A3 and 300B. Here the grid is also wound around two vertical supports, but in this case it is flat (although often curved outwards slightly around the cathode). Like the cylindrical structure, this lends itself readily to theoretical analysis.
Figure 5c is a compromise between the two. The grid and the plate are flat, but the cathode is a small cylinder. This is a common construction for small tubes such as the 12Ax7 family or the 6SN7. Its behavior is very close to the truly flat structure of Figure 5b. There are many variations on this; for example, the plates are often elliptical rather than rectangular as shown. Similarly, the cathode may be elliptical. Since the characteristics depend directly on the geometry, this results in less consistent operation and is therefore undesirable for audio.
The challenge for the tube designer is to get the grid as close as possible to the cathode. As we will see later, this is the principal factor affecting the mutual conductance of the tube, and for most purposes higher mutual conductance means better. However, there are problems. First of all, the slight variation in grid-cathode spacing inevitable due to manufacturing tolerances leads to indistinct characteristics, since different parts of the tube are behaving differently. Indeed, if the spacing is too close then there is a risk that the two may even come into contact, with disastrous results. Secondly, if the spacing is much less than the distance between the grid wires then the tube performance becomes difficult to predict and may even get worse. This in turn led to pressure to wind the grids from ever-finer wire at an ever-tighter pitch. Miniature tubes typically use grid wire that is less than one-thousandth of an inch thick, with a grid-cathode spacing at its closest of little more than this.
Most of the metal inside a tube is nickel, which has a number of desirable properties. First, it has a high melting point. Copper, for example, would be unsuitable because it would soften and distort. Secondly, it does not absorb a lot of gas onto its surface (the technical term for this is “adsorption”). Such adsorbed gas is gradually released into the vacuum of the tube, and it is important to minimize it. An exception to the use of nickel is the heater (or filament) which is generally made of tungsten because of the high operating temperature.
The plate is generally blackened, because a black surface radiates heat much more effectively than a lighter one. The only way that heat can be removed is by radiation, and it is this that sets the limit on the power dissipation of a tube. Power tubes have plates that include large radiating surfaces having nothing to do with their electrical function, and even small-signal tubes often have wings or corrugations that serve the same purpose.
Because of the very close clearances, and also because of the impact of tube geometry on electrical characteristics, it is vital to maintain the dimensions to very close tolerances. The metal wires that support the various electrodes are generally held apart by spacers made from mica, which is an excellent heat-resistant insulator that is also very vacuum-friendly and easy to work to precise dimensions.
The heater is inside the cathode. Since the cathode itself is small, there is little room for it and even less room for electrical insulation. A thin layer of alumina (aluminium oxide) is normally used, but at the high temperatures involved this is not a perfect insulator, and if the voltage difference is too high then it can break down altogether. This is why all tubes have a rating for the maximum voltage difference between the heater and the cathode, typically 100-200V. Exceeding this may result in insulation breakdown, and of course failure. Even at lower voltages, there is some resistive and especially capacitative coupling between the heater and the cathode.
It is a very instructive exercise to take a tube to pieces and to measure the internal dimensions. Don’t do this with a new Western Electric 300B, but take a defunct octal tube such as 6SN7 (the miniature tubes are too fiddly), break the envelope by wrapping it well in a cloth and tapping it with a hammer. (Be careful doing this, and be careful as well with the shards of broken glass afterwards. Remember that some parts are mildly poisonous, and don’t even think about doing it with transmitting tubes which can contain highly toxic beryllium). You can then gradually dismantle the electrode structure, seeing how it is put together. You will find, for example, that many tubes have tiny metal plates welded to the tops of the grid supports, whose job is to get rid of the heat generated on the grid due to grid current. (In this case of the 6AS7G, these are comparable in size to the plates themselves).
4. Space Charge and Current Flow
If plate voltage is applied to a cold tube, then an electric field is set up between the plate and the cathode. The voltage in this field increases linearly in the distance between the two electrodes. This is shown by the red line in Figure 6. Once the cathode is heated up and starts emitting, the electrons themselves alter the situation, because the electron carries a negative charge. Although this is very small (1.602·10-19 Coulombs), there are a lot of them, and the emitted electrons themselves contribute to the electric field in the space between the electrodes, modifying it from the straight line shown in red in Figure 6. The green line shows the effect using a simple analysis, ignoring initial electron velocity (Child's Law), while the blue line shows the true state of affairs when initial electron velocity is considered.
The term space charge is used to refer to the charge due to the electrons occupying the inter-electrode space. The space charge plays a crucial role in controlling the flow of current, which is at the heart of the controlled amplifying action of the vacuum tube. There are several myths and misunderstandings concerning space charge. It is often implied that it exists only in the immediate region of the cathode, and there are references to the “space charge region”. It is true that the space charge in this region has particular importance, as we shall see later, but it is present in the whole inter-electrode space. Another source of confusion is the “space charge tubes” which were developed in the 1950s, for use in car radios. To allow them to operate effectively on the 12V supply in the car, without needing a higher voltage, they used an unusual configuration of grids to limit the reduction in plate current due to the space charge. These are now of interest only to collectors.
Figure 6: Charge distribution between cathode and plate
The Child-Langmuir Law
The number of electrons depends on the current flow - the higher the current, the greater the number of electrons and therefore the greater the charge. Since the cathode feels the influence of the plate through the negatively-charged electrons between them, the increasing current reduces the attractive force of the plate until the two reach a balance. At this point of balance, the effective field at the surface of the cathode is reduced to zero. Moving away from the cathode, the electrons accelerate. The accelerating electrons spread out, just like cars on a freeway moving away from a traffic jam, reducing the space charge so that the field increases. This is illustrated by the green line in Figure 6, which shows what happens inside a tube when current is flowing under normal operating conditions (under this analysis).
The mathematics of this is beyond the scope of this article, but it is relatively straightforward to obtain the Child-Langmuir law (sometimes referred to simply as Child’s law), as it applies to a diode:
where: V = plate voltage
A = cathode area
d = distance between cathode and plate (cm)
This is the well-known 3/2 power law of the relationship between current and voltage. Note also the bottom of the equation: the current drops with the square of inter-electrode distance. This is why designers need to keep the physical dimensions of the tube as small as possible.
This equation (and the green line in Figure 6) is based on a simplification, that all of the electrons are emitted with zero residual energy. We have already seen that in fact they are emitted with a distribution of energy, and later on we look at the effect this has.
This law strictly applies only to infinite flat electrodes. In reality of course the electrodes are finite, but in a practical tube the effect of this is relatively minor. A similar equation applies to cylindrical electrodes, which still involves the 3/2 power of voltage, and in fact it can be shown that any electrode configuration will involve this factor.
One implication of this law is that the current flowing depends only on the strength of the field due to the plate in the immediate vicinity of the cathode. This field must be just strong enough to counter exactly the space charge set up by the resulting current. The field elsewhere serves to accelerate the electron flow but plays no role in determining its magnitude.
Saturation
The current emitted by the cathode under normal conditions is far in excess of the current that passes to the plate. The emission of an oxide cathode is upwards of 0.5A/cm2. Even for a power tube, the ratio between the emitted current and the operating current is at least 10. The maximum plate current rating that appears in the specification for a tube is based mainly on considerations of heating, and in pulse operation can safely be exceeded (as it was in TV sweep circuits, for example).
If all of the emitted electrons do pass to the plate, then the operation is completely different. The plate current becomes practically independent of the plate voltage. In a triode, the grid no longer has any control over the current. This mode of operation is called thermal saturation (or just saturation), in contrast to the normal mode which is sometimes called space-charge limited. Under these conditions the full effect of the field due to the plate is felt directly at the cathode, since the space charge (which does still exist) is no longer strong enough to counter it. In fact, the current does increase a little with plate voltage, especially with oxide cathodes, because the increasing field strength can pull more electrons out of the surface of the cathode. This is called the Schottky effect. It must be stressed though that in audio, no practical vacuum tube circuit operates in saturation, even for short periods. (There are uses in other areas. One is the noise diode, which is deliberately operated with a low heater voltage so that emission is low, and saturation occurs continuously. Another is in high-current pulse switching).
5. The Triode Field and Operation
Static Field
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:
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).
where: dgp = distance from grid to plate
a = distance between grid wires
rg = 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 (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 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. 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.
Figure 7: Triode equipotentials
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 Child’s Law, but we also need to know the value to use for the cathode-plate distance. A good approximation is given by:
where: deq = equivalent effective distance of diode plate
dcg = distance from cathode to grid
dgp = 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 Child’s Law gives the complete equation for cathode current under given conditions:
or
In this formula, everything except the grid and plate voltages is determined by the geometry of the tube’s construction, and is therefore a constant for a given tube. The factor P in the equation is called the perveance of the tube, and is the measure of the current it will pass under given conditions. It is just all of the terms from the first equation, outside the voltage term, gathered up into one variable. 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"
A triode is described essentially by three well-known so-called constants:
Voltage amplification (µ) As described above, the factor by which the grid reduces the effect of the plate voltage
Mutual conductance (Gm) 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 (rp) 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
In fact, only two of these characteristics are required, since the three are connected by the relation:
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.
Figure 9: Changes of characteristics "constants" with plate current
It can be seen that both Gm and rp vary a great deal. In fact, this follows from Child’s Law. With a little calculus (differentiating the formula with respect to Vg) and some algebra, we arrive at the formula:
In other words, the value of Gm increases with the cube root of the plate current, and hence the value of rp decreases with the cube root of the plate current. Since Gm 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 Gm and increase rp. For the 6SN7 shown, Gm is 3.2mA/V at 16mA, but drops to 1mA/V when operating at 1.5mA. As in this case, Gm typically drops faster than the formula predicts, particularly at low currents and high negative grid voltages.
The curves also show that µ is not really constant, either. Over the range 5 – 1 mA, it drops from 20 to 15, i.e. by about 25% - and this is a particularly linear tube in this regard. Later, miniature tubes show a steady drop over the whole operating range. The explanation for this is not so simple, and indeed the classic tube texts do not give any help. 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. It can be shown that the parts which are still conducting have a lower value of µ. (This phenomenon is called inselbildung, German for "island effect"). Another reason, particularly important at low currents, is that the value of µ 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 µ regions conducting. In consequence, the average value of µ falls with the reducing current, explaining the distinct tailing off at low currents.
6. Initial Electron Velocity
The derivation of Child’s 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 Child’s 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 formulae are shown below.
where: dvc = distance from physical cathode to virtual cathode
Ip = plate current
T = cathode temperature (typically around 1050°K)
I0 = total emitted current
Vm = potential at virtual cathode
Vp = plate voltage
dcp = distance from cathode to plate
A = cathode area
It can be seen that evaluation of these formulae requires knowledge of the emitted current from the cathode. This is a hard quantity to measure, and in addition 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 key 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 subsantial 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. In fact at very low currents (say under 1mA for small-sgnal tubes) the transfer characteristic, starts to have an exponential nature. In audio terms, its behaviour starts to be more like a junction semiconductor (for the same underlying physical reasons), producing larger amounts of higher-harmonic, which is to say unpleasant, distortion.
These formulae also 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 increased energy of the emitted electrons (due to the higher temperature) provides an additional reason why the plate current will increase. 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 0.3mA, 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 (dvc) 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:
(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 cathode’s emission during current peaks. This is a myth; these electrons have a very short lifetime before they return to the cathode, and correspond to less than 1nS of emission.
6. Noise
Basics
All electronic, and indeed electrical, devices produce noise. The most fundamental, and unavoidable, form of noise is called thermal noise or sometimes shot noise. It occurs because the flow of electricity is actually due to individual electrons. Even though the number of electrons is huge, there are still statistically predictable fluctuations, just like traffic on the highway. Because it is random, this noise occurs equally at all frequencies. The noise voltage developed by a simple conductor or resistor is given by:
where: enoise = noise voltage
k = Boltzmann’s constant, 1.38·10-23
T = temperature, in ºK
R = resistance
Df = bandwidth
For example a 1kW resistor at room temperature (approx. 300ºK), across the audio bandwidth from 0-20kHz, produces a noise voltage of about 0.6µV. When dealing with very small signals, such as a received radio signal or a phono pickup, this is a significant amount of noise.
Vacuum tubes produce thermal noise. They also produce noise in two other ways, called flicker noise and separation noise.