Continuation of Arc Discharges / Plasma
Physics 
This
section is going to focus on some of the more practical characteristics
of arc discharges, as opposed to the theoretical material covered in
the first section. One of the most important characteristics of the arc
discharge is its VI curve. This curve describes how the arc voltage
changes with respect to the arc current, and has many important
implications.
Fig 1: VI Curve
of a Particular Arc [1]
These
curves were determined by Hertha Ayrton for arcs between carbon
electrodes with the spark gap distance as a parameter and are
particular to her setup. Therefore, they are not valid for different
electrode materials, geometries, and spacings. They do, however,
illustrate some important principles that can be applied to more
general scenarios.
An interesting
characteristic to start with is the relatively flat
portion of the VI curve at higher currents. From a practical
perspective, this can greatly simplify design and analysis since the
arc voltage is relatively insensitive to changes in current. For
example, in designing a switching arc power supply, a much larger
output current ripple might be acceptable and therefore much smaller
and less expensive magnetics could be used. I haven't been able to find
much information on VI curves that extend to hundreds or thousands of
Amps, but the experience I have seems to indicate that the arc voltage
remains steady (within perhaps +/ 10 V) well into the 10s and 100s of
thousands of amps. At some point I'll have my own experimental data to
confirm or contradict this, and I'll be sure to post it when it's
available.
But what should we make
of this flat region? How do we explain it in
the context of the physical phenomena discussed earlier? The most
intuitive place to start is probably with the anode and cathode voltage
drop regions, which you may remember are two invisibly thin regions at
the electrode surfaces where the arc attaches, and which exhibit a
consistent voltage drop (on the order of 10  20 V or so at both
electrodes) which is a function of the electrode material. The cathode
drop region is particularly important, since the positive ion
bombardment at its surface creates the heat that frees electrons which
in turn flow through the discharge. The simple explanation is that
increasing the arc current increases the thermal power delivered to the
cathode surface, and this in turn generates more free electrons and
later, more free positive ions. With additional charge carriers
available, an increased arc current can be supported given the same arc
voltage. In this way, the arc is selfsustaining.
It's worth noting that
since the anode and cathode drop regions are so
thin, they have very little surface area through which they can conduct
heat to the surrounding air. Therefore, most of the energy consumed by
the drop regions (arc current times anode or cathode drop) goes into
thermal heating of the electrode. Part of this energy goes to releasing
electrons from the cathode surface, most of it is either radiated or
conducted away from the arc attachment region.
That addresses the
relatively constant voltage region associated with
large currents, but what of the region where arc voltage increases
quickly with decreasing arc current? For an explanation to this, we can
look again to magnetic constriction of the arc. You may recall that at
high arc currents a strong magnetic field is created which exerts an
inward force on the flowing charges and opposed thermal expansion of
the gases, confining the discharge to a small cross section. At low
arc currents a relatively small magnetic field is created, which exerts
very little inward force, which does little to oppose thermal expansion
of the gases. This expansion results in an arc which is much cooler,
and consequently is less completely ionized. This reduction in charge
carriers results in lower conductivity of the positive column, and an
increase in arc voltage.
Fig. 2: Low Current Arc
(~30mA)
Incidentally, the effect
of a cooler arc on conductance and arc voltage
is also relevant when energy is extracted from the arc in an
application such as heating another gas. In such an application heat
from the positive column would be transfered out of the arc, resulting
in lower temperature and lower conductivity, with the ultimate effect
of increasing the arc voltage for a given current. Another way to think
about this is as an energy balance. For a given current a certain
amount of power is required to maintain electron generation at the
cathode, replace heat lost to radiation and conduction, etc. This
requires a certain arc voltage to deliver the required power (arc
current time arc voltage.) If additional power is taken out of the arc
to perform some other task, then the arc voltage must increase in order
to supply that additional power. OK, back to the VI curve...
The VI curve is also an
important consideration when starting an arc.
Consider a circuit consisting of a number of ideal voltage supplies
connected to two electrodes. As shown in Fig. 3.
Fig. 3: Supply and
Electrode Circuit
Suppose the VI curve has
already been established for these
electrodes, and is graphed with the VI characteristics of the power
supplies (constant voltages.)
Fig. 4: Electrode and
Power Supply VI
Characteristics
Note that the arc VI
curve extends upwards along the Y axis at zero
current until it meets the curve which resembles those shown in Fig. 1.
In this case, the supply lines for the three supplies shown (the
constant voltage lines representing ideal supplies) each intersect the
load line (arc VI curve) twice, once at zero current and again at some
positive current. In this case, all three supplies can only reach the
zero current point, as none of them are capable of generating a high
enough voltage to cause a current to flow. Clearly, this indicates that
the supplies shown in Fig. 4 are not capable of causing the spark gap
breakdown which is necessary to start the arc.
Now, let's modify the
circuit from Fig. 3 to include some resistance is
series with the supply.
Fig. 5: Modified Circuit
with Resistance
If the supply voltage is
higher than the gap breakdown voltage, there
is now one intersection between the source and load lines, which
represents a stable operating point which the supply is able to reach.
Fig. 6: Modified Source and Load Lines
This is critically
important when designing an ignition circuit, since
some ignition circuits have very high source impedance and therefore
can only supply very limited current. An ignition circuit may be able
to cause a breakdown between the electrodes, but if it is unable to
supply enough current to cause the arc voltage to fall below the
voltage
of the supply, the arc will extinguish when the ignition circuit is
turned off.
Fig. 7: A) Improperly
Matched Ignition Circuit and Supply B) Properly
Matched Ignition and Supply
Fig.
7a illustrates a mismatched ignition and supply. Due to the ignition
circuit's high source impedance, it can only supply a small amount of
current to the arc, and the arc voltage never falls low enough for the
supply to be able to provide current to the arc. In Fig. 7b the lower
source impedance enables the ignition circuit to operate below the
source line allowing the supply to provide current to the arc. Note
that the supply and arc line will intersect at another point far off
the right side of the chart, which will ultimately be the stable
operating point of the circuit.
References
[1] J. D. Cobine, Gaseous
Conductors, New York:
McGrawHill Book Company,
Inc., pp.
203
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