My Electric Engine - Plasma Physics

A Brief Introduction to Arc Discharges / Plasma Physics

Depending on how sophisticated of a design is needed, this material can be relatively simple or full blown rocket science. So as not to scare anyone away, I'll start with the simple and easy to understand material and work on from there.

First, let's define some terms.

Neutral Atom - This is an atom which is electrically neutral, and has the same number of protons (+1 charge) and electrons (-1 charge.) At room temperature most gasses are neutral, or have bonded with other atoms to form neutral molecules.
Ionized Atom - An atom which receives a certain amount of energy may have one or more electrons knocked out of its orbit thus ionizing it. This results in a net positive charge on the atom. An atom that looses one electron is singly ionized, two electrons is doubly ionized, and so on.
Ion - An ion referred to an atom or molecule which has either lost or gained an electron, and as a result is electrically charged. For the purposes of this discussion, negative ions (an extra electron) are typically not a significant contributor to the processes described here. Ions can be formed by collisions with other particles, strong electric fields, or radiation.
Ionized Gas - A gas in which a significant fraction of its atoms are ionized is an ionized gas.
Plasma - Plasma is a gas which is ionized to the point that it exhibits significant electrical conductivity and is characterized by high temperatures.
Arc Discharge - This refers to an electrical current that flows through an ionized gas / plasma. The current must be large enough to self-sustain the ionized gas through which it flows. It also commonly referred to as an electrical spark.

Let's start with something simple like an electric spark. If you take two wires and put a high enough voltage across them, a spark will jump between the two.

Fig. 1: Electric Spark

But how high of a voltage is required to get a spark across a certain gap? It turns out that this isn't exactly the question we should be asking, instead we need to know the electric field strength across the gap. An electric spark will occur if the electric field strength exceeds a certain value. The electric field is measured in volts per meter, and is computed by dividing the voltage across the gap by the distance of the gap. So, for a gap of 1cm with a voltage of 100V, the electric field strength is 10,000 V/m. An electric field of approximately 3,000,000 V/m is required to cause a spark, so this 100V across 1cm would not result in a spark - but 30,000 V across 1cm or 3,000V across 0.1cm would.

Now we know how to get a spark, but why is it that a spark only forms if the electric field strength is very high? Well, let's look at how an electric field effects charged particles first. It turns out that an electrically charged particle in an electric field will feel a force equal to

Eq 1: Force on a Charged Particle in an Electric Field

where F is the force in Newtons, q is the charge in Coulombs, and E is the electric field in volts per meter. Also, from physics we know that a force on a mass will cause an acceleration as related by

Eq. 2: Relationship Between Force, Mass, and Acceleration

where F is the force in Newtons, m is the mass in kilograms, and a is the acceleration in meters per second per second. Accordingly, placing an electric charge in an electric field will cause a force on the charge resulting in an acceleration in the direction of the electric field. Let's imagine first an electron is placed in an electric field of varying intensity.

Fig. 2: Electron in a Varying Electric Field

Let's start with a voltage of 0 Volts across the gap. This results in an electric field of 0 V/m, and substituting this value into equation 1 results in no force applied to the electron, and it therefor will not move. Applying a small voltage of 10mV will result in an electric field of 1 V/m. If we were to follow the substitution through equation 1 and 2 we would find that the electron accelerates at some rate towards the left. Increasing the electric field by a factor of 10 results in the electron accelerating 10 times more quickly, and so on. Keep in mind that an electron moving at a certain velocity has some kinetic energy, as given by

Eq. 3: Kinetic Energy of a Moving Mass

Overall, applying a stronger electric field results in any electrically charged particles in the field accelerating more quickly, and therefore possessing more kinetic energy.

Now, let's get back to how this relates to creating a spark in a gas across some gap. If we were to look at a gas on an atomic level we would see mostly neutral molecules and some few electrons. If no electric field is applied there will be no net movement of the electrons, but at very weak electric fields the electrons will begin to migrate as shown. [1]

Fig. 3: Electrons Begin to Migrate Across Gap

In weak electric fields (i.e. low applied voltages) only a very few electrons will make it all the way across the gap as most will collide with neutral gas molecules. Increasing the field strength will result in more free electrons making it across the gap, until at a certain voltage all of the free electrons will cross the gap. At this point, increasing the voltage further will not result in more current because all available free electrons are already crossing the gap. This sort of discharge is called the First Townsend Discharge region, and can carry only minuscule currents up to a certain saturation current value. The current characteristic for this type of discharge is shown below.

Fig. 4: Current Flow Vs. Applied E Field for 1st Townsend Discharge

As the applied electric field is increased, the electrons will increase in speed and energy. At some point they will acquire enough energy to knock additional electrons off of neutral molecules that they collide with as they cross the gap. These additional electrons will also be accelerated across the gap and the current will begin to increase again. This sort of discharge is called the Second Townsend Discharge region, and is also limited to very small currents. [2]

Fig. 5: Electrons Ionize Molecules Resulting in Additional Current

Through the Second Townsend Discharge region the positive ions created by electron collisions will move slowly in the opposite direction of the electrons toward the cathode, their lower velocity due to their higher mass. In the Second Townsend Discharge region these ions do not contribute significantly to the processes described so far. Up to this point the current that the gap can carry is a strong function of the number of naturally occurring free electrons in the gap. Since the presences of these free electrons is in no way related to the electric field, first and second Townsend discharges are essentially externally fueled and are therefore not self-sustaining.

At some higher electric field strength the positive ions flowing toward the cathode being to play a critical role. This occurs when the electric field is strong enough to give the positive ions sufficient energy that when the collide with the cathode they deposit enough energy to free electrons from the cathode surface. These free electrons are then swept towards the anode, creating electron and positive ion pairs which then repeat this process. [3]

Fig. 6: Self-Sustaining Discharge Creates its Own Electrons

The critical process here is the creation of electrons at the cathode's surface. At some field strength this process becomes efficient enough to supply all of the electrons crossing the gap, removing any dependency on outside sources of current carriers. This is the critical point at which the discharge quickly transitions to an arc discharge, or a spark.

By observing the voltage across an arc discharge we can identify three key regions. Much of the voltage drop occurs across two invisibly thin regions directly adjoining the anode and cathode, which are referred to as the anode drop region and cathode drop region respectively. The positive column occupies the remainder of the gap. [4]

Fig. 7: Voltage Distribution as Function of Distance Into Spark

For a given current through an arc discharge, the power dissipated in the cathode drop region (arc current times cathode drop voltage) serves to heat the cathode and cause it to emit electrons, the power dissipated in the anode drop region (arch current times anode drop voltage) goes to heating the anode by electron bombardment (but serves no useful purpose), and the power dissipated in the positive column maintains the electron and ion concentrations that support the arc current.

References

[1] J. D. Cobine, Gaseous Conductors, New York: McGraw-Hill Book Company, Inc., pp. 143

[2] J. D. Cobine, Gaseous Conductors, New York: McGraw-Hill Book Company, Inc., pp. 144

[3] J. D. Cobine, Gaseous Conductors, New York: McGraw-Hill Book Company, Inc., pp. 213

[4] J. D. Cobine, Gaseous Conductors, New York: McGraw-Hill Book Company, Inc., pp. 299

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