Thermionic Tubes
Thermionic tubes are the devices that pioneered electronics.  They originated in the early 20th Century and are still in common use to this day.  You don't need to look far to find them in use today, they are used in most televisions for the picture tube and Microwave ovens use a Magnetron to generate the radiation for cooking.  X-Rays are still generated using tubes for medical imaging right up to the Cat scan machines.  The transmitters in radio and television stations still use valves for the transmitters and most musicians still prefer the sound of a valve amplifier.  For high power stuff, valves still hold on to their place.  Physically, transistors are more robust but electrically, valves are still more forgiving.  The origin of the valve stems back to when Edison invented the first light globe.  He added an electrode to one of his prototypes and noticed that a voltage was generated on the electrode despite the vaccuum between the filament and electrode.  His conclusion was that electrons were emitted by the filament.  This was known as the Edison Effect. Over the years, as more was learned, this led to the creation of the valve.  The text that follows is the best explanation of how they work that I can come up with given my very basic and probably flawed knowledge of valves.

The basic valve works by the emission of electrons from an electrode known as the Cathode.  The cathode releases electrons due to a high amount of energy provided by a heat source.  The excess energy provided by the heat source allows the electrons to overcome their attraction to their nuclei and form a cloud around the cathode.  There are two methods of providing the heating, Direct and Indirect.  Direct heating involves having the heater filament provide the supply of electrons and therefore becomes part of the circuit.  Indirect heating has the heater mounted inside a tubular cathode which supplies the electrons and is insulated from the heater. The cathode is often coated with with chemicals which increase the cathode's emission.  The cathode is then mounted inside of a larger hollow electrode known as the Anode.  For the electron cloud to be created, the entire area must be in a vacuum.
In the above diagrams, the picture on the left is of a directly heated cathode with the heater represented by the red line through the centre.  The diagram on the right is of an indirectly heated cathode with the electrically isolated heater in the centre.

When electricity is applied to the heater, electrons are emitted by the cathode and flow to the anode.  If the polarity is reversed and the supply of electrons is applied to the anode, no electrons can be emitted due to the lack of heating and therefore, no current can flow to the cathode.  This simplest of configurations is known as a DIODE.
In this diagram, you are looking from the top down and the picture on the left is the diode forward biased with the electrons flowing from the cathode at the centre to the anode on the outside.  The picture on the right is a reverse baised diode showing the electrons on the anode but no electrons are flowing to the cathode.
The name "Valve" is derived from the fact that the current flows in one direction only.  The name "Diode" comes from the fact that it contains two electrodes.
If we added a third electrode, it would become a TRIODE.
The field around the cathode is known as the "Space Charge" and extends outwards towards the anode.  As it spreads outwards, the charge becomes weaker and the pull of the anode has less effect.  If we placed this third electrode between the cathode and anode, it would have the effect of increasing or decreasing the space charge due to it's attraction or repulsion of the charge.  Placing it closer to the cathode maximizes it's influence on the field and therefore causes a significantly higher increase to the charge transferred to the anode.  Of course, since the third electrode is between the anode and cathode, it will block the path so it must be made out of a mesh so that electrons can pass through it and reach the anode.  The change in the flow of electrons between the cathode and anode far exceeds the small potential required on the grid to control it, therefore amplification occurs.
This third electrode used is known as the "Control Grid".
As the grid goes more negative, it repels the electrons emitted by the cathode and thus reduces the emission.  This reduces the space charge between anode and cathode and reduces the current at the anode.
As the grid goes more positive, it attracts the electrons emitted by the cathode and thus increases the emission.  This increases the space charge between the anode and cathode causing the current flowing to the anode to increase.
(keep in mind that electrons flow from negative to positive)
Because positive charges attract electrons, current will flow into the control grid.  This is bad!!  For this reason, the control grid is usually biased negatively so that it works by repelling the field and thus does not attract current.  the signal is mixed into the negative bias voltage so that a stronger signal will counteract some of the negative charge causing it to repel less and thus still create the larger potential at the anode.
This diagram shows a triode with the control grid just outside of the cathode.  The picture at the left shows the valve heavily negative biased and the space charge being repelled and thus not conducting nuch current to the anode.  The picture to the right is less negatively biased and thus the spage charge is much higher and more current flows to the anode.

To help prevent oscillation at higher frequencies, a fourth electrode was later added between the control grid and anode which acts as a screen.  The "Screen Grid" is given a positive DC potential so that it doesn't counteract the pull of the anode but is grounded to the cathode by a capacitor resulting in a ground potential to higher frequency capacitively generated currents which may otherwise induce current in the control grid and thus turn the amplifier into an oscillator. The screen grid, like the control grid, is mesh electrode.  Valves employing this fourth electrode are known as "TETRODES". 

Tetrodes have the side effect of higher electron velocities due to the positive DC potential on the screen grid increasing the pull on the electrons.  This higher velocity leads to secondary emisson of electrons 'bouncing' off the anode.  Because the screen grid is given a positive DC potential, it will attract these electrons and at low signal levels, the potential on the anode can become less positive than the screen grid causing the screen grid to attract the secondary emission electrons and cause a severe reduction in the current at the anode.  This has the effect of making the valves response non-linear with a sharp reduction in amplification occuring between the level of control grid potential where secondary emissions begin through to the level where the anode potential is high enough to overpower the screen grids pull. Due to the non linear response, tetrodes aren't very usefull for amplification but are very well suited to being used in oscillators.

To eliminate the problems caused by secondary emission, a third grid was added between the screen grid and the anode which was known as the "Suppressor Grid" and was connected to the cathode.  This grid allows the electrons to flow through from the cathode and control grid due to the pull of the screen grids positive potential generating the high velocity but when secondary emissions occur from the anode, the suppressor grid repels them back to the anode preventing non-linearity experienced with tetrodes.  Valves employing this fifth grid are known as "PENTODES"

There were also types of valves which employed various techniques to improve on these previous types of valves.  By aligning the screen grid wires in the 'shadow' of the control grid, a larger volume of electrons are able to flow which increases the efficiency. By using special plates to concentrate the electron flow into a high electron density beam, you can eliminate secondary emission, thus creating a simillar effect to having a suppressor grid but without needing one.  Such valves are known as "BEAM TETRODES" and "BEAM PENTODES" and are well outside of the scope of my knowledge.  I have never actuall worked with valves at all but am a collector. When possible, I will try and post pictures of some of the rarer tubes in  my collection.

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