A tetrode is an electronic device having four active electrodes. The term most commonly applies to a two-grid vacuum tube. It has the three electrodes of a triode and an additional screen grid which significantly changes its behaviour.
The grid nearest the cathode is the "control grid"; the voltage applied to it causes the anode current to vary. In normal operation, with a resistive load, this varying current will result in varying (AC) voltage measured at the anode. With proper biasing, this voltage will be an amplified (but inverted) version of the AC voltage applied to the control grid, thus the tetrode can provide voltage gain.
The second grid, called "screen grid" or sometimes "shield grid", provides a screening effect, isolating the control grid from the anode, reducing the parasitic capacitance between the two. This helps to suppress unwanted oscillation, and to reduce an undesirable effect in triodes called the "Miller effect", where the gain of the tube causes a feedback effect which increases the apparent capacitance of the tube's grid, limiting the tube's high-frequency gain. In normal operation the screen grid is connected to a positive voltage, and bypassed to the cathode with a capacitor. This shields the grid from the anode, reducing Miller capacitance between those two electrodes to a very low level and improving the tube's gain at high frequencies. When the tetrode was introduced, a typical triode had an input capacitance of about 5 pF, but the screen grid reduced this capacitance to about 0.01 pF. 
As the screen grid is positively charged, it collects electrons, which causes current to flow in the screen grid circuit. This uses power and heats the screen grid; if the screen heats up enough it can melt and destroy the tube. There are two sources of electrons collected by the screen grid—in addition to the electrons emitted by the cathode, the screen grid can also collect secondary electrons ejected from the anode by the impact of the energetic primary electrons. Secondary emission can increase enough to decrease the anode current, since a single primary electron can eject more than one secondary electron. The reduction in anode current is because the external anode current (through the connection pin) is due to the cathode-to-anode current minus the secondary emission current. This can give the tetrode valve a distinctive negative resistance characteristic, sometimes called "tetrode kink". This is usually undesirable, although it can be exploited as in the dynatron oscillator. The secondary emission can be suppressed by adding a suppressor grid, making a pentode, or beam plates to make a beam tetrode/kinkless tetrode.
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