| First the arc plasma is a resistor, resistance is dictated by the ionization level (conductivity is proportional to the density of ions/free electron pairs), the mobility of mainly electrons (what is their mean free path length, so what distance they can freely travel without any collision; the ions are so heavy they do not move that fast, so their contribution to the conductivity is small; but yet it still follows the conductivity of electrons, as each ion has "its own" free electron somewhere in the plasma) and the geometry of the discharge (cross section and length).
However the ionization level, as well as to some extend the free electron path are not constant, but vary according to pressure, but as well as the balance between ions creation (fast electron splits an atom into an electron and ion) and decay (when an electron meets ion at low speed, it tends to jump back into the free orbit and make a neutral atom from it again). The rate the electron/ion pairs are generated is then proportional to the number of existing electrons and to the probability an electron strikes with sufficient energy to cause the split, so in the anode column section, an exponential function of the electric field accelerating mainly the electrons. So their generation is proportional to the current flowing.
And once some amount of ion/electron pair are formed, they exhibit a parameter what could be called a halflife - a time it takes for half of the electron/ion pairs to recombine. It depends on the energy of the electrons (so speed, so temperature), so on the power so current density and the radiation energy losses.
Normally, in a steady state, there is always a balance so the amount of generated ions is equal to the amount of ions/electrond just recombining back to a neutral atom. The mechanism behind is rather simple: If there were excess of generated ions over the recombined ones, the ionization goes up, so does the conductivity of tye plasma, which causes the voltage, so the electrical field to drop. But smaller electrical field generates less ions, now matching the recombination, so balance is restored. Given all other parameters constant, the equilibrium always means the same electrical field, so ultimately (when integrated along the arc length) the same voltage drop, regardless of the current you are feeding in. This is the main reason why arc voltage tends to be very little, if all, depende t on the current.
Now the generation, as well as recombination, are steered by the velocities, so energies within the involved components. In other words their temperature. Highger temperature means ion generation is more likely and recombination is less likely. That means at higher plasma temperature the equilibrium field, so voltage, tends to become lower. And because higher temperature is usually caused by higher power dissipation, so by the higher current, we have the arc voltage dropping down when current increases. Viola, one mechanism responsible for a "negative dynamic resistance" effect.
Plus at really high ionization levels, so at high currents, the plasma may gradually be running out of the neutral atoms, so the equilibrium shifts to higher fields, so voltages, as when there are less atoms to split, you have to reach higher probability and that is by accelerating the electrons faster.
So this effect tends to maintain the positive dynamic resistance, saturating the conductivity.
Now the overall anode column is then the mix of both of these effects together. At lower currents (term current density and electrical field terms would be more appropriate, but if the geometry is fixed, it translates into a current and voltage drop), when there is enough atoms to be split, the negative resistance effect is dominant, hence the voltage is decreasing with the current.
Then asthe current increases, the conductivity saturation gradually eats away from the negative resistance, till the voltage becomes flat and then starts to rise back at really high currents.
So far all was described as a static equilibrium. But when things change in time, the inertia of the ionization starts to play a significant role, so once the current changes, initially the voltage follows the Ohms law and changes with the current. Only then with some time constant (1's till 100's us) the ionization change restores the equilibrium for the new current. So you see an overshoot in voltage. This inertia makes the arc behaving differently at different frequencies:
At low frequencies, the ionization density is able to follow the current, so the arc voltage becomes a piece wise constant (just the sign follows the sign of the current). But at high frequencies, the ionisation density can not respond to the fast changes, so is almost constant, so the plasma behaves like a nice linear resistor, with the resistance settling so the equilibrium is maintained over the long time.
This all was just the anode column. The main part of a luminous discharge, but not the only one.
What remains are the electrodes, namely the cathode. Now as the current is flowing, the electrons are moving towards anode, becoming absorbed there. For the plasma to remain macroscopically neutral, something has to replenish the electrons back to the plasma and that is the role of the cathode.
The cathode has to liberste the electrons from its material and that needs an electrical field to pull them out and the electrons being already agitated, so somehow hot, so the sum of those forces overcomes the force of the electrode material atoms to keep them back (characterized by the energy barrier). Or being kicked out by some ion hitting the cathode at a speed.
Now if the cathode is warmer, less electrical field is required. Because the electrical fieldforms another voltage drop (a cathode fall), which does not contribute to the electrons exciting atoms, so it does not radiate anything, it just becomes a power loss which just heats up the cathode (by accelerating the ions which then are bombarding it, making the crystal structure to shake, so warming it up).
So once the cathode heats up, lower field is sufficient to rip off the electrons, so less heat is generated. Till the cathode being so hot the electrons flow out freely, with no cathode drop. So we have another system, looking for an equilibrium: Cathode being too cold means high drop, that means higher dissipation there, so cathode warmup, so lower cathode fall, so lower the dissipation till it settles so just enough heat is generated to maintain the temperature. Now the heat is proportional to the current, so higher current means lower voltge drop to maintain the temperature.
Because this cathode voltage drop is in series with the anode column plasma, we have another contributor to the negative dynamic resistance of the discharge. But the time constant of this is more related to the electrode thermal inertia, usually being slower than even the typical mains frequency.
And still we assumed the pressure haven't been changing yet, that makes the thing way more complicated...
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