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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 4, JULY/AUGUST 2001
Near-Cathode Phenomena in HID Lamps
Mikhail S. Benilov, Member, IEEE
Abstract—A brief review is given of the present understanding of
cathode phenomena in arc discharges. A model of a near-cathode
plasma region of high-pressure arcs with hot cathodes is described.
Possible ways of integration of the model into numerical codes describing a lamp on the whole are discussed. An approach is suggested allowing one to estimate the upper limit of the cathode temperature and the lower limit of the temperature inside a cathode
spot without detailed calculations of a temperature distribution inside the cathode. Solutions describing a diffuse and spot modes of
current transfer to a thin thermionic cathode are presented and
found to compare favorably with available experimental data.
Index Terms—Arc cathodes, high-intensity discharge, high-pressure arcs, lamps.
I. INTRODUCTION
A
NECESSARY condition to achievement of high performance of high-intensity discharge (HID) lamps is a longenough life of electrodes, typically of the order of ten thousand
hours. Thus, a proper design of electrodes is of crucial importance. This applies primarily to cathodes, since anode effects
are believed to be less critical to performance. The design of
electrodes of HID lamps could be facilitated if the physics of
near-electrode phenomena were well understood and an appropriate calculation model developed. In addition, an adequate calculation model of near-electrode phenomena is a very important
constituent of any model of the lamp on the whole.
Near-electrode phenomena in high-pressure arc discharges
and, specifically, in HID lamps have been studied for many
years. Unfortunately, near-electrode regions represent a very
difficult object for experimental investigation, in the first place
due to their small dimensions. Theoretical investigation is hindered by the multiplicity and diversity of processes involved. A
considerable progress in understanding cathode phenomena in
high-pressure arcs has been achieved in recent years. It is the
aim of this paper to present this understanding. Note that advances have also been achieved in recent years in understanding
physics of the anode phenomena (e.g., [1]–[7]), however, anode
phenomena are beyond the scope of this paper because of lack
of space.
Paper MSDAD-S 00–34, presented at the 2000 Industry Applications Society
Annual Meeting, Rome, Italy, October 8–12, and approved for publication in the
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Production and Application of Light Committee of the IEEE Industry Applications Society. Manuscript submitted for review October 15, 2000 and released for publication March
28, 2001. This work was supported by FEDER and CITMA.
The author is with the Departamento de Física, Universidade da Madeira,
9000 Funchal, Portugal (e-mail: benilov@uma.pt).
Publisher Item Identifier S 0093-9994(01)05913-8.
II. CONCEPTS
NEAR-CATHODE PHENOMENA
ARC DISCHARGES
OF
IN
Concepts of near-cathode phenomena in arc discharges
may be divided into two groups. First, there are works in
which plasma-cathode interaction is considered as a collective
phenomenon occurring on length scales of the order from tens
of micrometers and higher and on time scales of the order
from milliseconds and higher. Reviews of early works in which
such an approach was used can be found in [8]–[13]; models
specifically designed for cathodes of HID lamps are presented
in [14] and [15] (see also [16], in which the model of [14] is reconstructed). Second, there are works in which plasma-cathode
interaction is considered as a sequence of individual events
(termed by different authors as microexplosions, or microspots,
or fragments) occurring on extremely small length and time
scales of the orders of micrometers and nanoseconds, respectively; see, e.g., the reviews of [17] and [18]. While models
considered in the framework of the first approach are stationary
or quasi-stationary, models considered in the framework of the
second approach are essentially nonstationary. Mechanisms of
electron emission and of cathode erosion usually considered
in the framework of the first approach are thermionic or
thermo-field emission and evaporation of the cathode material,
while mechanisms considered in the framework of the second
approach are explosive and/or field emission and evaporation
and/or explosive erosion.
During the last decade, experimental evidence has been collected confirming the existence of fragments (microspots) in
discharges on cold cathodes and, thus, supporting the second
point of view insofar as cold cathodes are concerned; see, e.g.,
[18] and references therein. No unambiguous evidence of such
type has been obtained for hot cathodes. On the other hand, there
is a general experimental observation that variation of the work
function and of the boiling temperature of the cathode material
does not affect strongly characteristics of the cathode part of
an arc on a cold cathode and does affect characteristics of the
cathode part of an arc on a hot cathode (for example, a decrease
of the work function causes a decrease of the cathode surface
temperature). While the former conforms to the conclusion that
the mechanisms considered in the framework of the second approach play a decisive role for cold cathodes, the latter suggests
that dominating mechanisms for hot cathodes are those considered in the framework of the first approach.
Cathodes of a HID lamp are hot,except fora short period of time
after ignition. Therefore, the arc–cathode interaction will be considered in this paper in the framework of the first above-described
approach, i.e., as a (quasi-)stationary collective phenomenon
dominated by thermionic or thermo-field emission.
Opinions expressed by different authors within such a theoretical approach still vary very strongly even with regard to such
0093–9994/01$10.00 © 2001 IEEE
BENILOV: NEAR-CATHODE PHENOMENA IN HID LAMPS
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TABLE I
STRUCTURE OF THE NEAR-CATHODE REGION
a fundamental question as to what is an effect of space charge
in the near-cathode region: while some workers believe that this
effect is minor (e.g., [19] and references therein), others consider that processes related to the space-charge sheath may play
a substantial or even a decisive role, in particular in heating the
cathode.
Recently direct measurements were performed [20] of the
near-cathode voltage drop under conditions typical for a HID
lamp, combined with thermal measurements [21]. It was shown
that the near-cathode voltage drop may be much higher than
most workers previously believed (up to 40 V). Obviously, this
finding promotes the second one of the two above-mentioned
theoretical points of view. Furthermore, the measurements [20],
[21] which refer to the diffuse mode of current transfer to the
cathode are in good agreement with recent theoretical works
[22]–[25], in which the space-charge sheath is of central importance. It was shown [23], [39], [40] that such-type models also
result naturally in an explanation of different modes of current
transfer (diffuse and spot modes) and give results for a transition
between the modes which (qualitatively) agree with the experiment. In this review, the near-cathode phenomena in high-pressure arc discharges under conditions typical for HID lamps are
discussed along the lines of the model [22]. Note that this model
has been tested in different environments [22]–[24], [26]; a similar model was employed in [25].
III. NEAR-CATHODE PLASMA REGION OF A HIGH-PRESSURE
ARC WITH A HOT CATHODE
The term near-electrode region, when applied to a high-pressure arc discharge, refers conventionally to a nonequilibrium
plasma region separating the electrode surface from the bulk
plasma in which local thermodynamic equilibrium (LTE) holds.
In other words, this is a plasma region adjacent to the electrode
surface where deviations from LTE are localized. Deviations of
three types are the most important: a deviation of the electron
from the heavy-particle temperature , a vitemperature
olation of ionization equilibrium (i.e., a deviation of the elecfrom the equilibrium density
defined
tron density
by the Saha equation), and a violation of quasi-neutrality, i.e.,
and . These deinequality of electron and ion densities
viations may be characterized by the following length scales,
respectively: the length of electron energy relaxation; the ionization length; the Debye length. Estimates show [22] that these
scales under conditions typical for high-pressure arcs are of the
m,
m,
m, respectively. Since these
order of
scales are essentially different, the near-cathode region may be
divided into three zones: a space-charge sheath, an ionization
layer, and a layer of thermal relaxation. This structure is illustrated by Table I.
From the point of view of calculation of the plasma–cathode
interaction, the most important regions are the ionization layer,
in which the ion flux to the cathode surface is formed, and the
space-charge sheath, in which ions going to the cathode and
electrons emitted from the cathode are accelerated. One can see
that the thickness of the sheath and of the ionization layer is
much smaller than typical dimensions of the cathode and than
the radius of (macro)spots. It follows that the current transfer
across the ionization layer and the space-charge sheath is locally
one-dimensional and governed by local values of the surface
temperature
and of the voltage applied to the ionization
layer and the sheath. Note that the voltage drop in the layer of
thermal relaxation is usually of the order of tenths of volt, which
is much smaller than the voltage drop in the ionization layer and
can be effectively considered as the
the sheath. Therefore,
local voltage drop in the near-cathode layer.
The main features of the model [22] for the calculation of
ionization layer and the sheath are as follows. The space-charge
sheath is considered as collisionless for ions. (Currently, work
is in progress in order to remove this limitation and to render
the model uniformly valid for collision-free to collision-dominated sheaths.) Space charge of the plasma is constituted by ions
and plasma electrons, a contribution of the emitted electrons is
minor. Plasma electrons in the sheath obey the Boltzmann distribution at a constant . Basic equations are a kinetic equation describing the motion of ions and the Poisson equation. A
boundary condition at the sheath edge is the Bohm criterion.
In the ionization layer, the plasma is considered as quasi-neutral. Electrons in the ionization layer obey the Boltzmann distribution at a constant electron temperature which is the same as
in the sheath. One of basic equations is the equation of motion
of the ion fluid accounting for ion inertia, ion pressure gradient,
electric field force, friction force due to elastic collisions of ions
with neutral particles, and friction force due to ionization of neutral particles. Another basic equation describes balance of the
electron energy in the ionization layer, the sources of the electron energy being the flux of the energy brought in the layer by
the emitted electrons accelerated in the space-charge sheath and
the work of the electric field over the electrons inside the layer
and the sinks being the flux of the energy carried away by the
electrons leaving the ionization layer for the sheath and for the
bulk plasma and losses for ionization.
Boundary condition on the plasma side of the ionization layer
is the condition of ionization equilibrium, boundary condition at
the edge of the space-charge sheath is the Bohm criterion.
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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 4, JULY/AUGUST 2001
The above equations are solved jointly with an equation describing electron emission from the cathode surface, which relates the density of the emission current to the cathode surand the electric field
at the cathode
face temperature
have been within the domain of
surface. Parameters
Schottky-amplified thermionic emission [27] in all the calculations performed until now, therefore the present version of
the model uses the Richardson–Schottky equation. Note that
in [28]–[30] the enhancement of electron emission due to individual electric fields of ions moving to the cathode was studied.
According to these works, the effect is likely to play a role if the
electric field at the cathode surface is of the order of 10 V/m
or higher while the ion density in front of the cathode is of the
order of 10 m or higher. Both values are rather high and it
is unclear whether they can be attained under conditions typical
for HID lamps.
In order to close the model, one should specify the heavyparticle temperature in the ionization layer. Calculations have
shown that variations of this temperature do not affect appreciably the solution. For definiteness, it will be assumed that this
temperature equals the cathode surface temperature .
Input parameters for the model are gas species, the pressure
of the plasma, work function of the cathode material, , and .
Output data are all local parameters of the near-cathode layer, in
particular, the local electron temperature, local densities of the
ion current, of the current of plasma electrons, of the emission
current, and local density of the net current from the plasma
to the cathode surface. One can determine also the local density
of the net energy flux from the plasma to the cathode surface,
which equals the difference between the energy brought to the
cathode surface by ions generated in the ionization layer and
accelerated in the space-charge sheath and by plasma electrons,
and between the energy removed from the cathode surface by
electron emission.
and
for
An example of calculated dependences
fixed is shown in Fig. 1. At low temperatures, function
is growing. As
increases, and reaches a maximum and
is
then steeply falls, eventually changing sign. Function
monotonically increasing.
is as follows.
A reason of nonmonotony of function
As the surface temperature grows, the electron emission current
also grows. The influx of the electron energy in the
density
ionization layer increases. At low temperatures, when the ionization degree of the plasma is low, the ion current density grows
. The flux of the energy deapproximately proportionally to
livered to the cathode surface by the ions also grows proportion. The same is true also for the emission cooling. The
ally to
ratio cooling/heating is approximately constant and the net heat
. At higher surface
flux density also grows proportionally to
temperatures, when the ionization degree approaches unity, the
increase of the ion current density slows down. The increase of
the ion heating also slows down, while the emission cooling con. The ratio cooling/heating
tinues to grow proportionally to
begins to increase. The increase of the net heat flux density bepasses through a maximum
comes slower. Subsequently,
and starts to decrease. In summary, a reason of
point
is the limitation of ion heating
nonmonotony of function
caused by full ionization of the plasma in the near-cathode layer.
Fig. 1. Dependence of the energy flux density (solid lines) and of the current
density (broken lines) from Ar plasma to a W cathode for various values of the
voltage drop. p = 1 bar. (1: U = 10 V; 2: 11.13 V; 3: 12 V.)
The range of values of
in which
is comparable to
unity is rather narrow, which is a consequence of the Arrhenius
character of the processes involved.
is
Effect of variation of the voltage drop on function
. A reason is that the
much stronger than that on function
ion current density, which provides a dominating contribution
to the cathode heating, depends on much stronger than the
emission current, which gives a dominating contribution to the
net current density.
Data plotted in Fig. 1 were calculated for a cathode made of
(pure) tungsten. In Fig. 2, results are shown for a cathode made
of thoriated tungsten, which is characterized by a substantially
lower work function than pure tungsten. One can see that the enand in the case
ergy flux and current densities for the same
of thoriated tungsten are substantially higher than those in the
case of pure tungsten. The maximum of the dependence
for thoriated tungsten is reached at lower surface temperatures
than for pure tungsten.
IV. INTEGRATING THE MODEL OF THE NEAR-CATHODE PLASMA
REGION INTO NUMERICAL MODELS OF A LAMP
At present, a numerical model of an arc column can be developed by means of some or other computational fluid dynamics (CFD) code, such as the commercially available package
FLUENT. Equations of heat conduction and of current continuity in the body of electrodes, describing distributions of temperature and electrostatic potential inside electrodes, also can
be solved without major difficulties, for example, by means of
commercially available software such as ANSYS. As a consequence, many groups around the world have codes describing
the arc column and the electrodes. However, physically reasonable results for the whole system arc electrodes cannot be obtained without modeling the near-cathode and near-anode regions. A possible way of integrating the above-described model
of the near-cathode region into such-type codes consists in describing the near-cathode region as an infinitesimally thin contact layer, heat generation and voltage drop in this layer being
BENILOV: NEAR-CATHODE PHENOMENA IN HID LAMPS
989
Fig. 2. Dependence of the energy flux density (solid lines) and of the current
density (broken lines) from Ar plasma to a thoriated-tungsten cathode for
1 bar. (1: U = 10 V; 2: 11.13 V; 3:
various values of the voltage drop. p
12 V.)
=
known functions of the local cathode surface temperature
and of the local current density .
The procedure is as follows. By means of the model described
in the preceding section, one can calculate, in the one-dimensional approximation, characteristics of the near-cathode layer
and . In particular, dependences
,
in function of
, and
should be found. (These dependences
can be found from data similar to those presented in Figs. 1 and
2.) It should be emphasized that these characteristics do not depend on the total arc current or on the shape of the cathode and
it is sufficient to calculate these characteristics only once for
every given combination of the cathode material, the plasmaproducing gas, and its pressure.
,
, and
have
After functions
been calculated, one can use these functions in order to formulate boundary conditions at the cathode surface for equations
describing the arc column and equations of heat conduction
and of current continuity in the cathode. Consider first the case
when the arc column is described by equations for a chemically
equilibrium two-temperature plasma, as is done in [24] and
[31]. Then, boundary conditions at the cathode surface are
(1)
(2)
and
(3)
,
, and
are distributions of the electron and
where
heavy-particle temperatures and of the electrostatic potential in
are distributions of the temperathe arc column, and
is the
ture and of the electrostatic potential in the cathode,
electrical conductivity of the arc plasma, and are thermal
and electrical conductivities of the cathode material, and is a
direction outside the cathode locally orthogonal to the surface.
Adding the usual zero boundary conditions at the cathode surface for the distribution of the plasma velocity in the arc column,
one obtains a closed description of the system arc cathode.
If the arc column is described in the LTE approximation
(by equations for a chemically equilibrium one-temperature
plasma), then first and second boundary conditions in (1) must
be replaced by a condition for the plasma temperature. If the
ionization degree of the plasma in the arc column is low and the
heavy-particle thermal conductivity exceeds substantially the
electron thermal conductivity, then the boundary condition in
question is
. If the ionization degree is not too low
and the heavy-particle thermal conductivity is much lower than
the electron thermal conductivity, then the boundary condition
. If the two thermal conductivities are
is
comparable, the boundary condition assumes an intermediate
form.
Note that the solution inside the cathode is coupled with the
solution in the arc column only through boundary condition (3).
The voltage drop in the layer of thermal relaxation and in the
LTE part of the near-cathode region is in most cases (in particular, in cases when the cathode operates in the diffuse mode) of
the order of 1 V, which is much smaller than the voltage drop in
the sheath and in the ionization layer. It follows that the plasma
potential does not vary appreciably along the cathode surface
in (3) may be considered as a constant. In such a sitand
uation, the solution inside the cathode is decoupled from the
solution in the arc column and may be found independently. In
other words, there is in principle no need to calculate the whole
system arc cathode simultaneously: one can first find a solution for the near-cathode layer, then a solution describing the
cathode, and finally a solution for the arc column.
More specifically, the procedure is as follows. After functions
and
have been determined, one solves the
heat conduction and current continuity equations in the cathode
body with boundary conditions (2) and (3) with a given constant
. Integrating the obtained current density distribuvalue of
tion over the cathode surface, one finds the arc current corre. This step
sponding to the considered value of parameter
until
can be repeated several times with different values of
a desired value of the arc current is reached. If the principal aim
of a study consists in a modeling of a cathodic part of the discharge, which is the case, e.g., in investigations dealing with
cathode erosion, the procedure of solution terminates here. If
this is not the case, one can calculate a solution for the arc
column, the boundary conditions for the respective equations
being (1) in which distributions along the surface of the temperature of the cathode and of the current density, obtained at the
previous step, are substituted.
The problem may be simplified further if one takes into account that the Joule heat production and the voltage drop in the
body of the cathode are negligible in most cases, in particular,
in cases when the cathode operates in the diffuse mode. Then,
the current continuity equation may be excluded from consideration and the potential may be set constant inside the cathode.
It follows that the voltage drop in the near-cathode layer, ,
takes the same value at all points of the cathode surface. In this
case, it is more convenient to calculate characteristics of the
rather than
.
near-cathode layer in function of
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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 4, JULY/AUGUST 2001
After dependences
,
, and
have
been found (again, it is sufficient to calculate these dependences
only once for every given combination of the cathode material, the plasma-producing gas and its pressure; see examples
in Figs. 1 and 2), one can determine the temperature distribution inside the cathode body and on the surface for a given .
This amounts to solving inside the cathode the heat conduction
equation supplemented with boundary condition
. After this problem has been solved, one can substitute the surface temperature distribution obtained as a part of
found at the previous
the solution into dependence
step, thus determining the current density distribution over the
cathode surface and the arc current corresponding to the considered value of parameter . This step can be repeated several
times with different values of until a desired value of the arc
current is reached.
V. UPPER LIMIT OF THE CATHODE TEMPERATURE AND LOWER
LIMIT OF THE SPOT TEMPERATURE
It follows from the above that the effect of processes in the
arc column on the cathode part of the discharge is in most
cases weak, which is a consequence of the fact that the energy
flux coming to the cathode is formed in a thin near-cathode
region. The temperature distribution inside the cathode body
and on the surface may in most cases be found by solving
inside the cathode the heat conduction equation supplemented
, where is the
with boundary condition
voltage drop in the near-cathode layer, a parameter which takes
the same value at all points of the current-collecting surface
of the cathode. In this section, the upper limit of the cathode
temperature and the lower limit of the temperature inside
cathode spots for a cathode operating in the spot mode will be
established on the basis of general properties of the above-mentioned problem governing the temperature distribution in the
cathode.
the range of values of the surface temDesignate by
perature corresponding to the falling section of function
for fixed (see Figs. 1 and 2), where
is the value of the surreaches the maxface temperature at which the function
is the value of the surface temperimum (for given) and
vanishes, i.e., the local emission cooling
ature at which
becomes equal to the local heating.
At surface temperatures exceeding , the cooling would exceed the heating and the local energy flux would be directed
from the cathode into the plasma. Introducing the heat flux poand making use of the maximum
tential
principle for harmonic functions, one can prove that the latter
cannot happen and the temperature of any point of the cathode
surface cannot exceed . A physical sense of this conclusion
can be
is quite clear: values of the temperature exceeding
maintained neither by the heat flux from the plasma (which is
), nor by the heat flux from adjacent points
negative for
of the cathode body (this flux could be originated only by points
of the surface where the heat flux from the plasma is positive,
however, heat cannot propagate from these points to a hotter
point).
The above-discussed upper limit of the temperature of the
cathode surface applies, within the assumptions made, to any
mode of current transfer to the cathode. In particular, it represents an upper estimate of the temperature inside cathode spots
for a cathode operating in the spot mode. One can estimate the
spot temperature also from below. This estimate may be established as follows. States on the growing section of the dependence of the function on the surface temperature for fixed
(Figs. 1 and 2) are potentially unstable: a local increase of the
temperature will result in an increase of the local energy flux
from the plasma, which will cause a new increase of the local
temperature, etc. One can assume therefore that the temperature
inside the spot should corresponds to the falling section of the
. Note that
function , i.e., should be within the range
this assumption conforms to results of numerical calculations
[32].
is usually 300–500 K, i.e.,
The temperature range
not very wide. It follows that the temperature inside a spot does
not change much and to the accuracy of about 200 K may be
.
estimated as
Thus, an estimate of the temperature of spots on thermionic
cathodes may be obtained without solving the heat conduction
equation in the cathode body: it is sufficient to find the dependence of the density of the net energy flux from the plasma to the
, for a given
cathode surface on the surface temperature,
. Since this dependence is strong, which is a consequence of
the Arrhenius character of thermionic emission, the temperature
of spots estimated in this way is governed mainly by the work
function of the cathode material; dependence on other parameters is relatively weak.
This conclusion conforms to the experimentally observed fact
that the spot temperature is virtually independent of the discharge current and of the cathode geometry; see, e.g., [33]–[37].
These works reveal that the spot temperature in atmosphericpressure argon arcs with cathodes made of thoriated or pure
tungsten with a current of the order of a few hundred amperes is
virtually independent of the exact value of the discharge current
and of the cathode geometry, being about 3.5 kK in the case of
a thoriated-tungsten cathode and about 1000 higher in the case
of a cathode made of pure tungsten.
In a sense, this feature is similar to the effect of normal current
density observed on cold glow cathodes: as the discharge current
increases the spot expands, local parameters in the spot being
virtually constant.
VI. DIFFUSE AND SPOT MODES ON A THIN CATHODE
A simple solution to the problem governing the temperature
distribution in the cathode body can be obtained for the following model [23], [39]: a cathode represents a right cylinder of
a cross section which is not necessarily circular; the current-collecting surface of the cathode is its top; the lateral surface is
insulated; the bottom is maintained at a fixed temperature by
external cooling. It was found that this model allows one to not
only investigate main features of diffuse and spot modes of current transfer to a cathode, but also to obtain results which agree
with experimental data obtained on thin cathodes (of a height
exceeding substantially the radius).
BENILOV: NEAR-CATHODE PHENOMENA IN HID LAMPS
991
Fig. 3. Current–voltage characteristics of the diffuse mode of current transfer
mm, p
: bar). Lines: theory.
to a cathode in the argon plasma (h
;
: experiment [20]. Solid line, : tungsten cathode. Broken line, :
thoriated-tungsten cathode. (Figure reprinted with permission from [41].)
+
= 24
= 26
+
Fig. 5. Current–voltage characteristics of various modes of current transfer to
mm, R
a circular-cylinder tungsten cathode in the argon plasma (h
mm, p
bar). (Figure reprinted with permission from [41].)
= 10
=1
Fig. 4. Surface temperature of a cathode operating in the diffuse mode in
the argon plasma. p
:
bar. Lines: theory. ; ; : experiment [21].
;
;
;
;
: W cathode. 3: thoriated-tungsten cathode. ; ; : h
mm. : h
mm. ;
h
mm. (Figure reprinted with permission
from [41].)
12+2
= 24
= 26
2 + : = 19
+2
132
= 29
The problem governing the temperature distribution in the
cathode body has in the considered geometry a one-dimensional
(here, axis is directed along the axis of the
solution
cylinder), in which varies linearly with . The temperature
at all points of the top of the cathode is the same, hence this
solution describes a diffuse mode.
A comparison of this solution with the recent experimental
results [20], [21] obtained in experiments with a model HID
lamp with a thermionic cathode is shown in Figs. 3 and 4 [38]
(here designates the height of the cathode). The cathode in the
experiments represented a circular cylinder of the radius of 0.75
mm. In calculations, the cathode radius was increased by 50%
in order to account for the current collected in the experiment
by a part of the lateral surface of the cathode adjacent to the top
surface. The agreement between the theory and the experiment
is good.
Apart from the one-dimensional solution, the considered
problem may have multidimensional solutions
which branch off from (or join) the one-dimensional solution
and describe various spot modes. Bifurcation analysis of the
=2
problem has been performed in [23] and [39]. An example of
current–voltage characteristics (CVCs) of various modes is
is the
shown in Fig. 5 [38] (here, is the arc current and
represents CVCs of the
radius of the cathode). The line
,
,
diffuse mode and has been calculated numerically.
, and
represent schematics of CVCs of, respectively,
the mode with a spot at the edge of (the top surface of) the
cathode, the mode with two spots at the edge, the mode with
a spot at the center, and the mode with a ring spot at the edge.
These lines have been computed in the vicinity of bifurcation
points (points , , and ) and drawn with the use of results
of a qualitative analysis outside this vicinity.
Qualitative conclusions [23] concerning a transition between
the diffuse mode and the first spot mode conform to available
experimental information [12], thus supporting the hypothesis
that the multiplicity of modes of current transfer to hot arc cathodes is related to nonuniqueness of multidimensional thermal
balance of a finite body heated by a nonlinear external energy
flux. A numerical treatment of multidimensional solutions has
been performed in [40], [41].
VII. CONCLUSIONS
Understanding of cathode phenomena in arc discharges has
considerably improved in recent years due to progress both in
experiment and in theory. A review of the present understanding
of cathode phenomena in high-pressure arc discharges on hot
thermionic cathodes has been given. The effect of processes in
the arc column on the cathode part of the discharge is in most
cases weak, which is a consequence of the fact that the energy
flux coming to the cathode is formed in a thin near-cathode
plasma region. A theoretical model of the near-cathode region
was described. The model can be used both by itself or as a part
of a code modeling the whole system arc electrodes. In particular, the model results in a simple estimate of the upper limit of
the cathode temperature and of the lower limit of the temperature inside a cathode spot. A simple solution for the temperature
distribution in the cathode body can be obtained in the case of
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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 4, JULY/AUGUST 2001
a thin cathode. Results obtained on a diffuse mode are in good
agreement with data [20], [21] of experiments on a high-pressure argon model lamp. Qualitative theoretical conclusions concerning a transition between the diffuse mode and the first spot
mode conform to available experimental information [12].
ACKNOWLEDGMENT
Portions of the work reported in this paper were performed
within activities of the projects Generation, Diagnostics and
Modeling of Reactive Plasmas and Nonlinear, Stochastic and
Complex Systems: Their Mathematical Theory and Application
of the program Praxis XXI (Portugal) and Fundamental Characterization of Thermionic Cathodes of BMBF (Germany).
REFERENCES
[1] H. Salihou, J. P. Guillot, M. Abbaoui, and A. Lefort, “Anode parameters
of short arcs at low current,” J. Phys. D, Appl. Phys., vol. 29, no. 11, pp.
2915–2921, 1996.
[2] G. A. Dyuzhev, N. K. Mitrofanov, and S. M. Shkol’nik, “Experimental
investigation of the anode region of a free-burning atmospheric-pressure
inert-gas arc—I: General characteristics of the discharge. Low-current
regime,” Sov. Phys.—Tech. Phys., vol. 42, no. 1, pp. 30–34, 1997.
[3] F. G. Baksht, G. A. Dyuzhev, N. K. Mitrofanov, and S. M. Shkol’nik,
“Experimental investigation of the anode region of a free-burning atmospheric-pressure inert-gas arc—II: Intermediate current regime—Multiple anode constriction,” Tech. Phys., vol. 42, no. 1, pp. 35–38, 1997.
[4] J. Jenista, J. V. R. Heberlein, and E. Pfender, “Numerical model of the
anode region of high-current electric arcs,” IEEE Trans. Plasma Sci.,
vol. 25, pp. 883–890, Oct. 1997.
[5] T. Amakawa, J. Jenista, J. V. R. Heberlein, and E. Pfender, “Anodeboundary-layer behavior in a transferred, high-intensity arc,” J. Phys.
D, Appl. Phys., vol. 31, no. 20, pp. 2826–2834, 1998.
[6] M. Tanaka, M. Ushio, and C. S. Wu, “One-dimensional analysis of the
anode boundary layer in free-burning argon arcs,” J. Phys. D, Appl.
Phys., vol. 32, no. 5, pp. 605–611, 1999.
, “Observations of the anode boundary layer in free-burning argon
[7]
arcs,” J. Phys. D, Appl. Phys., vol. 32, no. 8, pp. 906–912, 1999.
[8] G. A. Lyubimov and V. I. Rakhovskii, “The cathode spot of a vacuum
arc,” Sov. Phys.—Usp., vol. 21, no. 8, pp. 693–718, 1978.
[9] G. Ecker, “Theoretical aspects of the vacuum arc,” in Vacuum Arcs:
Theory and Application, J. M. Lafferty, Ed. New York: Wiley, 1980,
pp. 228–320.
[10] V. I. Rakhovsky, “State of the art of physical models of vacuum arc
cathode spots,” IEEE Trans. Plasma Sci., vol. PS-15, pp. 481–487, Oct.
1987.
[11] F. G. Baksht, V. G. Ivanov, and V. G. Yur’ev, “Electrode phenomena
in low-temperature plasma,” in Invited Papers XVIII ICPIG (Swansea,
Wales, July 1987), W. T. Williams, Ed. Bristol, U.K.: Adam Hilger,
1987, pp. 206–218.
[12] W. Neumann, The Mechanism of the Thermoemitting Arc
Cathode. Berlin, Germany: Akademie-Verlag, 1987.
[13] E. Hantzsche, “Theoretical models of the cathode spot in arc discharges,” Contrib. Plasma Phys., vol. 32, no. 2, pp. 109–169, 1992.
[14] J. F. Waymouth, “Analysis of cathode-spot behavior in high-pressure
discharge lamps,” J. Light Vis. Environ., vol. 6, no. 2, pp. 53–64, 1982.
[15] E. Fischer, “Modeling of low-power high-pressure discharge lamps,”
Philips J. Res., vol. 42, no. 1, pp. 58–85, 1987.
[16] G. G. Lister and H. G. Adler, “Modeling and diagnostics of HID lamp
electrodes,” in Proceedings of ICPIG XXIV (Warsaw, July 1999), P. Pisarczyk, T. Pisarczyk, and J. Wolowski, Eds. Warsaw, Poland: Inst. of
Plasma Physics and Laser Microfusion, 1999, vol. 1, pp. 229–230.
[17] G. A. Mesyats, “Ectons and their role in electrical discharges in vacuum
and gases,” in Invited Papers ICPIG XXIII (Tolouse, July 1997), M. C.
Bordage and A. Gleizes, Eds. Les Ulis, France: EDP Sciences, 1997,
pp. C4-93–C4-112.
[18] B. Jüttner, “Properties of arc cathode spots,” in Invited Papers ICPIG
XXIII (Tolouse, July 1997), M. C. Bordage and A. Gleizes, Eds. Les
Ulis, France: EDP Sciences, 1997, pp. C4-31–C4-45.
[19] J. J. Lowke, R. Morrow, and J. Haidar, “A simplified unified theory of
arcs and their electrodes,” J. Phys. D, Appl. Phys., vol. 30, no. 14, pp.
2033–2042, 1997.
[20] J. Luhmann, D. Nandelstädt, A. Barzik, and J. Mentel, “Measurements
of the cathode fall in a high pressure argon model lamp,” in Proceedings
of ICPIG XXIV (Warsaw, July 1999), P. Pisarczyk, T. Pisarczyk, and J.
Wolowski, Eds. Warsaw, Poland: Inst. of Plasma Physics and Laser
Microfusion, 1999, vol. 1, pp. 13–14.
[21] D. Nandelstädt, J. Luhmann, B. Micheldt, and J. Mentel, “Measuring
the temperature distribution at thermionic cathodes in an argon model
arc,” in Proceedings of the 8th International Symposium on the Science
and Technology of Light Sources (Greifswald, Germany, Sept. 1998),
G. Babucke, Ed. Greifswald, Germany: Inst. for Low-Temperature
Plasma Physics, 1998, pp. 350–351.
[22] M. S. Benilov and A. Marotta, “A model of the cathode region of atmospheric pressure arcs,” J. Phys. D, Appl. Phys., vol. 28, no. 9, pp.
1869–1882, 1995.
[23] M. S. Benilov, “Nonlinear surface heating of a plane sample and modes
of current transfer to hot arc cathodes,” Phys. Rev. E, Stat. Phys. Plasmas
Fluids relat. Interdiscip. Top., vol. 58, no. 5, pp. 6480–6494, 1998.
[24] J. Wendelstorf, “Two-temperature, two-dimensional modeling of
cathode-plasma interaction in electric arcs,” in Proceedings of ICPIG
XXIV (Warsaw, July 1999), P. Pisarczyk, T. Pisarczyk, and J. Wolowski,
Eds. Warsaw, Poland: Inst. of Plasma Physics and Laser Microfusion,
1999, vol. 2, pp. 227–228.
[25] H. Schmitz and K.-U. Riemann, “Consistent analysis of the boundary
layer of a Saha plasma,” in Proceedings of ICPIG XXIV (Warsaw, July
1999), P. Pisarczyk, T. Pisarczyk, and J. Wolowski, Eds. Warsaw,
Poland: Institute of Plasma Physics and Laser Microfusion, 1999, vol.
1, pp. 5–6.
[26] W. Hu, F. P. Dawson, and M. S. Benilov, “Modeling of cathode heating
in vortex-water-wall-high-pressure-argon-arc lamps,” in Proceedings
of the 8th International Symposium on the Science and Technology
of Light Sources (Greifswald, Germany, Sept. 1998), G. Babucke,
Ed. Greifswald, Germany: Institute for Low-Temperature Plasma
Physics, 1998, pp. 306–307.
[27] E. L. Murphy and R. H. Good, “Thermionic emission, field emission,
and the transition region,” Phys. Rev., vol. 102, no. 6, pp. 1464–1473,
1956.
[28] B. Y. Moizhes and V. A. Nemchinskii, “On anomalously high emission from a nonevaporating thermocathode in an arc discharge,” Sov.
Phys.—Tech. Phys., vol. 18, no. 12, pp. 2539–2547, 1974.
[29] R. Gayet, C. Harel, T. Josso, and H. Jouin, “A simple model for cathodic
electronic emission enhanced by low-energy ions in high-pressure arcs,”
J. Phys. D, Appl. Phys., vol. 29, no. 12, pp. 3063–3070, 1996.
[30] T. Josso, H. Jouin, C. Harel, and R. Gayet, “Enhancement of cathodic
electronic emission by slow positive ions in high-pressure arcs,” J. Phys.
D, Appl. Phys., vol. 31, no. 8, pp. 996–1008, 1998.
[31] K. Charrada, G. Zissis, and M. Aubes, “Two-temperature, two-dimensional fluid modeling of mercury plasma in high-pressure lamps,” J.
Phys. D, Appl. Phys., vol. 29, no. 9, pp. 2432–2438, 1996.
[32] M. S. Benilov, “Nonlinear heat structures and arc-discharge electrode
spots,” Phys. Rev. E, Stat. Phys. Plasmas Fluids relat. Interdiscip. Top.,
vol. 48, no. 1, pp. 506–515, 1993.
[33] M. Ushio, A. Sadek, and F. Matsuda, “Comparison of temperature and
work function measurements obtained with different GTA electrodes,”
Plasma Chem. Plasma Process., vol. 11, no. 1, pp. 81–101, 1991.
[34] J. Haidar and A. J. D. Farmer, “A method for the measurement of the
cathode surface temperature for a high-current free-burning arc,” Rev.
Sci. Instrum., vol. 64, no. 2, pp. 542–546, 1993.
, “Temperature measurements for high-current free-burning arcs in
[35]
nitrogen,” J. Phys. D, Appl. Phys., vol. 26, no. 7, pp. 1224–1229, 1993.
[36]
, “Effect of cathode shape on plasma temperature in high-current
free-burning arcs,” J. Phys. D, Appl. Phys., vol. 27, no. 3, pp. 550–560,
1994.
[37]
, “Surface-temperature measurements for tungsten-based cathodes
of high-current free-burning arcs,” J. Phys. D, Appl. Phys., vol. 28, no.
10, pp. 2089–2094, 1995.
[38] M. S. Benilov, “Theory of nonlinear surface heating,” Phys. Scr., vol.
T84, pp. 22–26, 2000.
[39] M. S. Benilov and N. V. Pisannaya, “Mathematical modeling of constricted current flow to an electrode with a thermal resistance,” Sov.
Phys.—Tech. Phys., vol. 33, no. 11, pp. 1260–1266, 1988.
[40] R. Bötticher and W. Bötticher, “Numerical modeling of arc attachment
to cathodes of high-intensity discharge lamps,” J. Phys. D, Appl. Phys.,
vol. 33, no. 4, pp. 367–374, 2000.
[41] S. Coulombe, “Arc-cathode attachment modes in high-pressure arcs,” in
Bull. Amer. Phys. Soc., 53rd Gaseous Electronics Conf., vol. 45, 2000,
p. 18.
BENILOV: NEAR-CATHODE PHENOMENA IN HID LAMPS
Mikhail S. Benilov (M’98) received the Diploma
and the C.Sc. (Ph.D.) degree in physics from the
Moscow Institute for Physics and Technology,
Moscow, U.S.S.R., and the Doctor of physical and
mathematical sciences degree from the Institute
for High Temperatures, U.S.S.R. Academy of
Sciences, Moscow, U.S.S.R., in 1974, 1978, and
1990, respectively.
After completing postgraduate courses at the
Moscow Institute for Physics and Technology and
at the Institute for Mechanics, Lomonosov Moscow
State University, in 1977, he joined the Institute for High Temperatures,
U.S.S.R. Academy of Sciences, where he led a group working in plasma and
nonlinear physics, numerical modeling, and fluid dynamics. In 1990, he was
awarded the Alexander von Humboldt Research Fellowship and spent two
years at the Ruhr-Universität Bochum, Germany, working on the theory of
near-electrode phenomena. Since 1993, he has been a Professor of Physics
at the University of Madeira, Funchal, Portugal. During 1996–1998, and
since June 2000, he has served as Chairman of the Department of Physics.
His research interests include plasma physics (in particular, near-electrode
phenomena and electrostatic probes), nonlinear physics, and fluid dynamics.
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