Download Pulsed currents carried by whistlers. VIII. Current disruptions and

Pulsed currents carried by whistlers. VIII. Current disruptions
and instabilities caused by plasma erosion
R. L. Stenzel and J. M. Urrutia
Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547
~Received 21 August 1996; accepted 15 October 1996!
In a large magnetized laboratory plasma (n.1012 cm23 , kT e >1 eV, B 0 >10 G, 1 m 3 2.5 m!, the
transient processes of switch-on currents to electrodes are investigated experimentally. The current
rise time lies between the ion and electron cyclotron periods ~electron magnetohydrodynamics!. The
initial current scales linearly with applied voltage and is not limited by the electron saturation
current of the positive electrode, but by the ion saturation current of the return electrode. The
collection of electrons in the flux tube of the positive electrode gives rise to a space charge electric
field, which expels the unmagnetized ions, erodes the density, and disrupts the current. Repeated
current oscillations arise from a feedback between current, density, and potential oscillations. The
dependence of the transient and unstable electrode currents on externally variable parameters is
investigated in the present paper. A companion paper @Urrutia and Stenzel, Phys. Plasmas 4, 36
~1997!# presents in situ measurements of plasma currents, plasma parameters, and microinstabilities.
These results are relevant to the physics of pulsed Langmuir probes, current collection from tethered
electrodes in space, and plasma erosion switches. © 1997 American Institute of Physics.
@S1070-664X~97!02801-2#
I. INTRODUCTION
Time-dependent plasma currents between electrodes in
magnetized plasmas arise in many applications in laboratory
and space plasmas. For example, the current-voltage characteristics of diagnostic Langmuir probes is often swept rapidly
in time so as to obtain time-resolved plasma parameters. A
well-known feature is a current ‘‘overshoot’’ which occurs
near the plasma potential for positive voltage ramps.1 It has
been explained by the inertia of ions in a sheath which
changes from ion-rich to electron-rich.2 Alternatively, the
rapid variation of the plasma potential relative to a constant
probe potential in rf discharges can drastically change the
current-voltage characteristics of the probe.3 In space plasmas, the rapid motion of satellite-borne electrodes leads to
time-dependent currents in the stationary magnetoplasma.4
If the time variation lies between the electron and ion
cyclotron periods, the physics of the transient electromagnetic processes is described by electron magnetohydrodynamics ~EMHD!.5 In this regime, detailed laboratory experiments have shown that the transient current is transported in
the whistler mode, forms force-free configurations, and behaves linearly up to field energy densities exceeding the particle energy density.6–12 Nonlinearities arise when the fields
change the plasma parameters. Electron heating is the fastest
process in collisional plasmas, occurring on the time scale of
electron-ion collisions. It locally increases the Spitzer conductivity and causes current filamentation.11,13
The next nonlinearity arises from the motion of the ions
which occurs on a time scale of an ion transit time across the
current channel. The associated nonlinear effects and instabilities are the topic of the present work. It describes the
transition from the transient current front to steady-state currents in weakly magnetized plasmas where the ions are mobile but unmagnetized ( n in . v ). For currents small compared to the electron saturation current, the behavior is
linear, i.e., current and voltage waveforms are identical step
26
Phys. Plasmas 4 (1), January 1997
functions. The current density exhibits helicity, J•B Þ 0,
even in steady-state. For electrode potentials exceeding the
plasma potential, the current waveform exhibits a transient
overshoot and a relaxation oscillation. The peak current is
not limited to the initial electron saturation current.
However, such large currents last only for the duration
of an ion transit time across the current channel. The collapse
of the transient current produces an overshoot effect similar
to that described earlier1 but of fundamentally different properties calling for a new explanation. For example, the duration of the overshoot is not an ion transit time across the
sheath but across the entire electrode. The height of the overshoot is so large that it cannot result from excess ions in a
sheath.2 The initial current is driven by an electric field that
is not confined to the Debye sheath but penetrates as a whistler wave field over many collisionless skin depths into the
plasma. For such time-dependent fields, the current-voltage
characteristic is governed by an ac Ohm’s law rather than dc
Langmuir probe theory. The large current collapses due to a
major density depletion in the flux tube of the electrode. The
density ‘‘erosion’’ produces essentially a plasma opening
switch.14 After the collapse of the transient current, the
smaller ‘‘steady-state’’ current can exhibit relaxation oscillations with a frequency approximately given by the ion sound
speed divided by the electrode size. This relaxation oscillation is not a ringing response to the transient current overshoot, but a genuine instability which also arises when the
voltage is turned on slowly, i.e., without initial current overshoot.
Steady-state oscillations generated by positively biased
probes in weakly magnetized discharge plasmas have been
observed earlier but without explaining the instability
mechanism.15 In strongly magnetized Q-machine plasmas,16
positively biased ‘‘buttons’’ can excite current-driven ion cyclotron
instabilities
which
have
been
studied
extensively.17–22 At larger bias voltages, the button modifies
1070-664X/97/4(1)/26/10/$10.00
© 1997 American Institute of Physics
FIG. 1. Experimental setup, diagnostics and basic plasma parameters.
the plasma production and losses and generates ‘‘potential
relaxation instabilities,’’ which modulate the button
current.23 However, in spite of 60 years of research on
probes and their use, there is surprisingly little information
available about the type of instability discussed in the present
work. Related current disruptions by plasma erosion and
double layer formation have been observed in magnetic reconnection experiments.24 They have also been studied in
low density plasmas without electromagnetic effects.25,26 Recently, their importance to electromagnetic tethers in space
has been pointed out.27,28 The present work describes many
new features of the EMHD current transient and instability
not reported earlier. The variety of measurements and parameter scalings contributes to the understanding of the nonlinear, transient current phenomena.
The paper is organized as follows: After describing in
Sec. II the experimental setup and measurement techniques,
the observed results are presented in the various subsections
of Sec. III. The conclusion, Sec. IV, points out the relevance
of the present findings to related observations and applications. While the first of the two companion papers focusses
on external current-voltage measurements, the second one29
describes internal measurements of plasma parameters,
fields, currents, and microinstabilities.
(Dr'1 cm, Dt'50 ns!. From Ampère’s law, the conduction
current density, J5¹3B/ m 0 @ e 0 ] E/ ] t, is calculated without making any assumptions about field symmetries or relying on ¹–B50. The measurement accuracy of B can be expressed as u ¹–Bu / u ¹3Bu .5%. The plasma parameters are
obtained from a small Langmuir probe ( p r 2 .2.6 mm2 ),
which is also movable in three dimensions. Typical Langmuir probe errors are ¹n/n.10%. Miniature coaxial probes
~0.1 mm diam 3 1 mm! are used to detect and correlate
fluctuations excited by current-driven instabilities.
II. EXPERIMENTAL ARRANGEMENT
III. EXPERIMENTAL RESULTS
The experiments are performed in a large laboratory
plasma device schematically shown in Fig. 1. A 1 m diam
3 2.5 m long plasma column of density n e .1012 cm23 ,
electron temperature kT e .2 eV, and argon gas pressure
p n .0.4 mTorr, is produced in a uniform axial magnetic field
(B 0 .30 G! with a pulsed dc discharge (V dis .55 V,
I dis .1200 A, t pulse .5 ms, t re p .1 s! using a large oxidecoated cathode.30 In the quiescent, uniform, current-free afterglow, switched plasma currents (t rise <1 m s, I<100 A!
are applied with disk electrodes (<2 cm diam! biased positively (V 0 <250 V! with respect to the end chamber wall.
The time-varying magnetic fields associated with the plasma
currents are measured with a triple magnetic probe, recording (B x , B y , B z ) versus time at a given position. By repeating the highly reproducible discharges and moving the probe
to . 15 000 positions in a three-dimensional ~3-D! volume,
the vector field B(r,t) is obtained with high resolution
A. Rapidly swept Langmuir probe traces
Phys. Plasmas, Vol. 4, No. 1, January 1997
FIG. 2. Overshoot features of rapidly swept Langmuir probe characteristics.
Slow voltage ramp, V(t) ~dashed line!, produces a small overshoot in the
current, I(t), near the plasma potential in a cold plasma ~lower trace!. In a
warm plasma ~upper trace!, no overshoot arises since it takes longer to
sweep out the retardation region. ~b! Faster voltage ramp produces a larger
current overshoot in the cold plasma of parameters as in ~a!. Timedependent peak current does not yield the electron saturation current.
We start with the familiar observation that a rapidly
swept voltage ramp applied to a Langmuir probe creates a
current overshoot near the plasma potential.1 Figure 2 shows
probe voltages and currents vs time. The former varies linearly in time and is shown for two different sweep rates
@ dV/dt.0.5 V/ m s in ~a! and dV/dt.2.5 V/ m s in ~b!#. The
currents correspond to plasmas at two different temperatures
but same densities. At the higher temperature (kT e .1.4 eV!,
the retardation region is swept out within Dt.10 m s, which
is sufficiently slow to produce no current overshoot. At the
lower temperature (kT e .0.2 eV!, the corresponding time is
much shorter (Dt.2 m s! and an overshoot is visible. The
current overshoot grows as the sweep time is decreased
@ Dt.1 m s in ~b!#, implying that the peak current is not the
electron saturation current. The duration of the overshoot
(Dt.1 m s! roughly corresponds to an ion transit time at the
R. L. Stenzel and J. M. Urrutia
27
FIG. 4. Time-resolved current-voltage characteristics for a 4 mm diam disk
electrode to which a step-function voltage is applied at t50. Insert shows
typical large-amplitude current waveform, I(t). During the collapse of the
current overshoot, the I(t)2V el (t) characteristics exhibits a negative differential resistance. At late times, a ‘‘normal’’ Langmuir probe characteristics
is observed but the saturation current is lower than the theoretical value,
I e,sat .0.52 A.
FIG. 3. Current, I(t), to a disk electrode produced by step-function voltages
of different amplitude, V 0 . At voltages below the plasma potential
(V 0 ,F pl .8 V!, current and voltage have the same waveforms. With increasing V 0 , a transient current overshoot and subsequent relaxation
oscillations develop. Insert shows peak overshoot current, I max vs V 0 . It
is not limited to the theoretical electron saturation current,
I e,sat 52 p r 2el ne(kT e /2p m e ) 1/2.24 A. The subsequent time-average current, ^ I(t) & , is smaller than I e,sat .
sound speed (c s .105 cm/s! across the size of the probe
(Dr.1 mm! rather than across the Debye sheath
(s.5l D .0.05 mm!. Thus, the overshoot phenomenon cannot be due to an adjustment of the sheath but is possibly
caused by a density perturbation on a scale of the probe
dimension. The saturation current observed after the collapse
of the initial current yields the locally perturbed plasma
properties rather than those in the absence of the probe.
A swept probe voltage makes it difficult to separate
time-dependent and voltage-dependent phenomena. An alternate approach to ramping the probe voltage is to apply a
step-function waveform @ V(t<0)50, V(t.0)5V 0
5 const#, record the current vs time, I(t), and to repeat the
experiment at different voltages V 0 . Figure 3 shows singleshot traces of I(t) for a relatively large disk electrode
(r el 51 cm.r ce .1.3 mm at 30 G, 1.4 eV! with surface
normal along B0 . Electrons are collected from both sides of
the disk. At voltages smaller than the plasma potential
(F pl .8 V!, the current has the same step-function waveform as the voltage. But as the voltage exceeds the plasma
potential, an initial current overshoot develops. The peak
current, I max , exhibits no saturation at the Langmuir limit,
I e,sat 52 p r 2el ne AkT e /2p m e . 24 A for the initial plasma parameters, and increases almost linearly with applied voltage
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Phys. Plasmas, Vol. 4, No. 1, January 1997
as shown in the insert of Fig. 3. The current rise is limited by
the circuit inductance to dI/dt.V/L.V 0 /2.8 m H
. 0.36V 0 A/Vm s. The duration of the current overshoot decreases with voltage but is always much larger than the voltage rise time (t rise .100 ns! or the ion plasma period
( f 21
pi .30 ns!, which excludes capacitive and sheath effects
as the cause for the overshoot. After the current has collapsed to a value smaller than I e,sat , it exhibits strong fluctuations which consist of repeated current spikes similar to
the first one but at a lower amplitude. The period of the
repeated current spikes (t re p .6 m s! corresponds roughly to
an ion acoustic transit time across the current channel of
diameter determined by that of the electrode.
Before going further into the physical processes of the
current disruption, it may be of interest to examine the
current-voltage (I-V) characteristics from which one usually
derives the plasma parameters. Figure 4 shows I-V curves
for a smaller probe (r el 52 mm! and finer voltage increments
(DV el .2 V! than in Fig. 3. At the time of the current maximum @ t.0.4 m s, trace ~a!#, the I-V characteristic exhibits no
saturation. This feature indicates that the electric field is not
initially limited to the Debye sheath but extends into the
plasma and drives currents obeying an Ohm’s law. The slope
of the I-V curves is also influenced by the inductance of the
wire leading to the probe. This will be examined further
below. During the current collapse, the I-V characteristic has
a region of negative differential resistance (dI/dV,0). This
can lead to possible instabilities with external circuit elements. Apparently ‘‘normal’’ Langmuir characteristic is only
observed after the collapse of the transient current and in the
absence of subsequent current spikes. However, the inferred
density is too low compared to the initial unperturbed value.
Averaging the time-dependent current does not yield the correct electron saturation current either. Thus, large planar
probes are of limited use for plasma diagnostics. The objecR. L. Stenzel and J. M. Urrutia
FIG. 5. Characteristic differences of the transient current overshoot and the
subsequent current oscillations. ~a! Step-function voltage applied to a onesided 2 cm diam disk electrode in an active discharge plasma of indicated
parameters. ~b! Single trace of electrode current showing both initial transient and subsequent relaxation oscillations. ~c! Ensemble average over
N520 repeated experiments shows that the transient overshoot is highly
reproducible while the current oscillations phase-mix due to variations in
amplitude and timing. ~d! Slowly rising voltage waveform with same peak
value (V 0 563 V! produces current waveform ~e! without transient but unstable relaxation oscillations. Oscillation period increases during growth due
to widening of the density-depleted current channel.
tive of the present work is to understand the mechanism of
the perturbations, to find conditions which minimize the errors for diagnostic applications, and to optimize the disruptions for switching applications.
In the following subsections, the dependence of the transient current on various externally variable parameters will
be discussed, such as voltage waveform, electron density and
temperature, magnetic field, ion species, neutral density,
electrode size, geometry, orientation with respect to B0 ,
properties of the external circuit, and role of the return electrode. While this information is highly useful, conclusive
evidence requires in situ measurements of plasma currents
and parameters, which are presented in the companion
paper.29
B. Current overshoot, disruptions, and relaxation
oscillations
First, we point out some important differences in the
properties of the first current overshoot and the subsequent
current spikes. The former is a transient phenomenon, the
latter is a current-driven instability. Characteristic features
are summarized in Fig. 5. A rapidly rising step function voltage @Fig. 5~a!# applied during the active discharge to a 2 cm
Phys. Plasmas, Vol. 4, No. 1, January 1997
diam, one-sided disk electrode produces both an initial current transient and subsequent relaxation oscillations as
shown in the single current trace of Fig. 5~b!. An ensemble
average over N520 repeated traces, ^ I(t) & N520 @Fig. 5~c!#,
shows that the initial current transient is highly reproducible
while the unstable current oscillations phase-mix due to
variations in amplitude and timing. The current oscillations
do not depend on the existence of the initial current overshoot. In Fig. 5~d!, the voltage rises slowly to the same dc
value as in Fig. 5~a!. In this case, the current @Fig. 5~e!# has
no transient overshoot, but the same current oscillations as in
Fig. 5~b! are observed to grow. The oscillation frequency
decreases as the amplitude increases. It is shown below that
the frequency depends on sound speed and radius of the
density-depleted current channel. Thus, the frequency pulling suggests that the depleted channel grows radially to an
asymptotic size given by the electrode radius. No frequency
pulling occurs when the current transient rapidly erodes the
density profile.
Under certain conditions, the current waveform exhibits
only a transient but no oscillations. This occurs ~i! at lower
voltages ~see Fig. 3, V 0 520 V!, indicating a threshold for
the current instability, ~ii! for different electrode sizes and
geometries as discussed below, and ~iii! in H1 -Ar1 plasmas
and in the late afterglow where kT e .kT i , suggesting that
damping of ion sound waves quenches the instability. The
latter point is demonstrated in Fig. 6. At an early afterglow
time @Fig. 6~a!, top trace, kT e .1 eV#, the current, I(t), exhibits both a transient overshoot and relaxation oscillations
while at a later time, i.e., lower electron temperature @Fig.
6~a!, lower trace, kT e .0.3 eV#, the oscillations are absent.
The loss of density @25% for t n 52n/( ] n/ ] t).2.5 ms# cannot explain the disappearance of the oscillations, since density variations by an order of magnitude at higher temperatures have no influence on the instability. In a pure argon
plasma at kT e >1 eV, the current relaxation oscillations are
present @Fig. 6~b!, top trace#, but they are absent in a mixture
of argon and hydrogen @Fig. 6~b!, bottom trace#. The addition of the light ions does not lower the electron temperature,
but can cause ion Landau damping of sound waves in
argon.31
C. Dependence on plasma parameters
The above described current waveforms have been observed over a wide range of magnetic fields (4,B 0 ,80 G!
and plasma densities (1010,n e ,1012 cm23 ) in both afterglow and active discharge plasmas. They are present in various gases ~H, He, Ar, Kr! over a wide range of pressures
(1024 , p,1023 Torr!. The influence of ion mass on the
current waveform has been investigated in krypton and argon
at the same neutral pressure. The time scales of the current
transient and the instability period are found to increase with
the square root of the ion mass ( v Ar / v Kr.1.4). The frequency is nearly independent of neutral density, plasma density and magnetic field, which is characteristic of an ion
acoustic oscillation.
A new physical effect arises at both high neutral densities and high electrode voltages which changes the current
waveform dramatically. Ionization, accompanied by excitaR. L. Stenzel and J. M. Urrutia
29
FIG. 7. Modification of transient currents by ionization. ~a! Current and
light emission at low neutral pressures in the absence of ionization. ~b! At
higher neutral pressures light emission indicates excitation/ionization. Electrons are energized at an anode double layer. Current rises when ion production exceeds ion expulsion by electric fields. Neutral pressure drop near
electrode causes quenching of ionization and current collapse.
FIG. 6. Suppression of the relaxation oscillations due to damping of ion
acoustic waves. ~a! Current, I(t), at different electron temperatures, controlled by the afterglow time. No oscillations are observed at low values of
T e /T i ~bottom trace!. ~b! Current, I(t), in an argon plasma ~top trace! and in
a hydrogen-argon plasma ~bottom trace! where current oscillations are suppressed by ion Landau damping.
tion, is clearly identified from light emission measurements
with a photodiode. Figure 7 presents a comparison of electrode current and light emission at two different neutral gas
pressures. At the lower pressure @Fig. 7~a!#, no light emission
is observed and the ensemble-averaged current exhibits the
typical transient and oscillations. At the higher pressure @Fig.
7~b!#, the observed light emission indicates ionization. The
current collapse of the initial transient is reversed and large
currents (I max .40 A! are collected. Probe measurements
near the electrode show that a potential double layer is
formed near the positive electrode, which accelerates electrons to energies exceeding the ionization energy. The current rises as long as the production of ions by ionization
exceeds their loss by acceleration in the electric field. At a
constant voltage, the current and light decay at t.30 m s
indicates a loss of ionization. The loss arises when the outflow of fast ions exceeds the inflow of slow neutrals such
that the gas pressure drops and the ionization is quenched.
Enhanced current collection by ionization is used for contactors of electrodynamic tethers.32,33 Anodic double layers with
ionization are well described in the literature.34–36
Figure 8 shows that the current relaxation oscillations
are also affected by ionization phenomena. While keeping
electron density and temperature approximately constant
(1.231012,n e ,1.331012 cm23 , 1.4,kT e ,1.6 eV!, an increase in neutral density is found to delay the onset of the
30
Phys. Plasmas, Vol. 4, No. 1, January 1997
secondary current pulses. From in situ density measurements
reported in the companion paper,29 it is shown that the current oscillations arise when large currents are flowing in a
density-depleted flux tube. Ionization compensates for the
density loss and delays the onset of the instability. Since
plasma production is proportional to the neutral density,37
one finds a linear increase of the delay time, Dt, between the
first and second current pulse and the gas pressure ~Fig. 8,
insert!.
FIG. 8. Dependence of current waveform, I(t), on neutral gas pressure,
p n . With increasing pressure, the onset of relaxation oscillations is delayed
since ionization refills the depleted current channel. Insert shows linear increase of delay time, Dt, between the first and second current pulse vs
neutral pressure. The delay results because plasma production increases linearly with pressure.
R. L. Stenzel and J. M. Urrutia
FIG. 9. Dependence of current waveform, I(t), on electrode size and orientation. ~a! Currents to three planar electrodes of different cross sections with
surface normals along B0 . Voltage and initial plasma parameters are constant. For electrodes smaller than an electron Larmor radius ~top trace!, the
narrow overshoot (Dt.0.15 m s! is followed by a dc current close to the
theoretical electron saturation current. The overshoot widens for larger electrodes, and is followed by oscillations whose period increases with electrode
radius. ~b! Voltage and current waveforms for a one-sided, 2 cm diam disk
electrode with surface normal along and across B0 . In the latter case, the
oscillations vanish while the transient current remains.
D. Dependence on electrode size and geometry
The electrode size and its orientation with respect to
B0 have a significant effect on the current waveform. Figure
9~a! shows the current waveforms for three electrodes of
different sizes. Their surface normals are along B0 , the same
step-function voltage is applied (V 0 51100 V, t rise <0.1
m s!, and the plasma parameters are identical (n e .1012
cm23 , kT e .1.3 eV, B 0 530 G!. The smallest electrode
(r.0.65 mm, 2 p r 2 .2.6 mm2 , top trace! whose size is less
than an electron Larmor radius (r ce .1.3 mm! exhibits only
a short initial current transient (I max .0.35 A, Dt.0.15
m s! followed by a dc current (I dc .80 mA! close to the
theoretical electron saturation current (I e,sat .83 mA!. However, for electrodes large compared to r ce ~middle and bottom traces!, the overshoot is followed by multiple current
peaks whose widths and repetition times increase with electrode radius. For the large electrode, the half width of the
current overshoot (Dt.5.6 m s for r52.5 cm! corresponds
approximately to an ion transit time across the electrode radius at a speed v i 5r/Dt.4.53105 cm/s or energy
1
2
2m i v i .4.1 eV. The initial current rise time is determined by
the applied voltage and the inductance of the 1.5 m long wire
connecting the pulse generator to the probe (L'2.8 m H!,
which limit the initial current rise to dI/dt<V/L537 A/m s
as observed. Figure 9~b! shows the effect of electrode orientation with respect to B0 . When the disk is oriented with its
Phys. Plasmas, Vol. 4, No. 1, January 1997
surface normal n̂'B0 , the current oscillations vanish but the
transient overshoot is almost as large and lasts as long as
when n̂i B0 . In the latter case, the current waveform is not
affected if n̂ is anti-parallel to B0 , i.e., whether the one-sided
collector faces toward the cathode or away from it. The ratio
of peak currents, I',max /I i ,max .0.84, varies only by 65%
when the magnetic field is varied by 600% ~10 G,B 0 ,60
G, not shown!. The current ratio decreases with increasing
electrode dimensions. The large cross-field current can neither be a Pedersen current @ I P5 p r 2 s' E' .60 mA at
E' .1 V/cm, s' . s i ( n / v ce ) 2 . 0.02 V 21 cm21 ], nor a parallel current to the small surface area projected along B0
(2rd.0.03p r 2 ). Direct measurements of the current density
near the electrode, shown in the companion paper,29 demonstrate that the cross-field current is an electron Hall current
@ s H. s' ( v ce / n )@ s' # produced by a space charge electric
field E'n̂, B0 in front of the electrode outside the sheath
(Ei n̂ at the electrode!. The duration of the transient current
is approximately given by an ion transit time across the electrode along E. Since the Hall electric field does not energize
electrons, no ionization phenomena are observed for identical conditions as in Fig. 7~b!.
Further detailed investigations of the current dependence
on electrode size and orientation with respect to B0 have
been performed with a probe of continuously variable cross
section. It consists of a rectangular tantalum plate ~3.2 cm
32.2 cm30.4 mm!, which can be retracted between two
insulating mica sheets so as to vary the double-sided collecting area in the range 0,A,14 cm2 . With the surface normal
along B0 and V 0 5 const, the measured current has been
normalized to the collector area and plotted as current density vs electrode area @Figs. 10~a! and 10~b!#. The ion current
density is found to be independent of surface area as expected from probe theories.2 Basic properties of the ion current are shown in Fig. 10~c!. Its temporal behavior exhibits
no nonlinearities or instabilities comparable to those observed at similar positive voltages ~see Fig. 3!. There is a
capacitive current transient, I5CdV/dt, which is associated
with the sheath and circuit capacitance (C.250 pF,
C sheath . e 0 A/5l D . 178 pF, C line .72 pF. The ion current
follows classical I-V characteristics.2 In earlier experiments
in the same device, a test electron beam has been used to
measure accurately the electron plasma frequency.38 It was
shown then that the ion saturation current yields, to within
62%, the same density when an empirical Bohm factor of
0.43 is used, i.e., I i,sat 50.43Ane(kT e /m i ) 1/2. Thus, under
the present conditions, switched ion currents are reliable for
time-resolved plasma diagnostics. Returning to the electron
currents, one observes that, for V 0 @kT e /e, the peak overshoot current density, J e,max @Fig. 10~a!#, and the ensembleaveraged current density obtained at a late time (30,t,50
m s!, ^ J e & @Fig. 10~b!#, decrease with area indicating that the
collection of large electron currents perturbs the plasma. The
most probable perturbation is a density loss, i.e., plasma erosion. This explanation is supported by direct measurements
discussed in the companion paper.29 For small areas, the average electron current density appears to converge to the
theoretical
electron
saturation
current
density,
J e,sat 5ne(kT e /2p m e ) 1/25 0.93(m e /m i ) 1/2J i,sat , indicated
R. L. Stenzel and J. M. Urrutia
31
FIG. 10. Comparison of plasma perturbations by electron and ion currents
for an electrode of variable area, A. Current densities, J5I/A, vs electrode
area for the ~a! peak transient current, ^ J e,max & , ~b! the theoretical, J e,sat ,
and measured, ^ J e,sat & , electron saturation currents, and the ion saturation
current, J i,sat . The observed loss of electron current density with increasing
area indicates a perturbation in the plasma density. ~c! Time variation of the
ion current, I i (t) ~insert!, shows no instabilities and the transients are caused
by sheath and cable capacitances, I5CdV/dt. The I-V characteristics follow theory indicating that ion current collection does not perturb the plasma
and is suited for time-resolved plasma diagnostics. The same appears to hold
for electron collection with electrodes smaller than the electron Larmor
radius.
in Fig. 10~b! by a dashed line. This convergence and the fact
that there are no fluctuations in the electron saturation current on probes smaller than the electron Larmor radius ~Fig.
9~a!, top trace! indicate that the I-V characteristics of small
electrodes appears to be useful for diagnostic purposes. For
large currents, the resistive voltage drop along the copper
wire between the pulse generator and the electrode has to be
taken into account (R wire .0.5 V). As the area increases, the
electrode voltage, V el 5V 0 2I max R wire , decreases and, since
I max } V el , the current density also decreases. After correcting for the resistive voltage drop, the current density,
J e,max (V 0 /V el ), displayed in Fig. 10~a!, still decreases
strongly with area. The voltage correction is negligible
(V 0 /V el ,1.03) for the smaller average electron current
( ^ I e & ,5 A!. Inductive voltage drops are absent at the current
maximum. The nonlinear scaling again suggests that the loss
of current density is caused by a modification of the plasma
parameters, e.g., a loss in density, which occurs on the time
scale of the overshoot.
Figures 11~a! and 11~b! present a comparison of the current collection along and across the magnetic field (B 0 550
32
Phys. Plasmas, Vol. 4, No. 1, January 1997
G! with the variable-area probe. The transient current is displayed vs time and electrode area in linear contour plots.
When the surface normal, n̂, is along B0 @Fig. 11~a!#, both
the peak current and the disruption time increase linearly
with surface area. The initial current (t,0.5 m s! is
inductance-limited, hence independent of electrode size provided A Þ 0. When the surface normal is across B0 @Fig.
11~b!#, the peak current is reduced and occurs earlier than in
the case n̂i B0 . Furthermore, when the adjustable electrode
length l (0,l,3.2 cm, across B0 ) exceeds its constant
width w (w52.2 cm, along B0 ), the peak current and the
disruption time do not further increase with electrode size. In
this case, the electrode becomes ‘‘magnetically’’ insulated.
Again, this property cannot be explained by resistive crossfield currents or edge currents but indicates a limitation in
the electric field, E, parallel to l, which controls the electron
E3B drift into the electrode. Considering the global circuit
@see insert in Fig. 11~b!#, the current waveform is determined
by a constant external inductance (L.2.8 m H! and a timedependent plasma resistance, R pl (t). The latter acts like an
opening switch, i.e., its rapid increase produces the current
disruption. From the circuit equation, the plasma resistance,
R pl (t)5(V 0 2LdI/dt)/I, is readily evaluated and plotted vs
time and electrode area for both n̂i B0 @Fig. 11~c!# and
n̂'B0 @Fig. 11~d!#. The logarithmically spaced contours ~6
dB/contour! indicate that in the former case the resistance
varies exponentially both in time, with scale
t 5R/(dR/dt).1 m s (A.4 cm2 ), and electrode area, with
scale a 5R/(dR/dA).21.2 cm2 (t.2 m s!. In both orientations, the initial resistance is comparably low (R pl .2 V at
A.4 cm2 ), but it does not further decrease when l.w for
n̂'B0 . The plasma resistance exceeds the resistance of the
copper wire of the external circuit. A more detailed analysis
of the circuit effects is given below.
E. Dependence on external circuit parameters
The inductive and resistive voltage drops in the external
circuit can cause a significant difference between the voltages at the generator and the electrode. By connecting a voltage sensor directly to the 2 cm diam disk electrode, the voltage, V el (t), has been recorded simultaneously with the
current, I(t), and both are shown in Fig. 12~a!. From the
schematic circuit diagram @Fig. 12~a!, insert#, one infers that
V 0 5IR1LdI/dt1V el where V 0 590 V is the open-loop
voltage or charging voltage of a large capacitor connected to
a transistor switch. The effective resistance and inductance
of the circuit are found from a linear least-square fit of
(V 0 2V el )/I vs (dI/dt)/I, which yields R.1.6 V, L.2.9
m H during the current rise. The main contribution comes
from the 1.5 m long tantalum wire between the generator and
the electrode. However, the internal resistance of the transistor switch becomes important at larger currents. The effect of
these circuit elements is to limit the short-circuit current
(I max 5V 0 /R.56 A! and its rise time (L/R.1.8 m s!, as
well as to produce large inductive open-loop voltages at the
electrode during current disruptions (V max .V 0 ). For larger
electrodes, hence larger currents, the inductive voltage,
LdI/dt, during the current disruption can far exceed the applied dc voltage.24 Alternatively, rather high open-loop voltR. L. Stenzel and J. M. Urrutia
FIG. 11. Contours of transient current, I, and plasma resistance, R pl 5V el /I, vs variable-electrode area and time for two different electrode orientations with
respect to B0 ~see inserted sketches!. ~a! Current, I(A,t), for surface normal n̂i B0 showing a linear increase in peak current and disruption time with electrode
area, A52lw @left scale as in ~c!#, or length, l ~right scale!. ~b! Current with n̂'B0 shows similar properties as in ~a! but the waveform becomes independent
of electrode size when l>2 cm .w. ~c! Plasma resistance @see equivalent circuit in ~b!# for surface normal n̂i B0 increases by over an order of magnitude at
the time of current disruption. The disruption timing depends on electrode size. The plasma resistance has the characteristics of an opening switch. ~d! Plasma
resistance for n̂'B0 shows similar behavior as in ~c! except when the resistance becomes essentially independent of electrode size.
ages are required in order to generate fast-rising, highcurrent pulses, e.g., V 0 >4500 V for I max 5150 A,
t rise .0.1 m s, L.3 m H.12
The ratio V(t)/I(t), plotted in Fig. 12~b!, describes the
impedance behavior of the electrode-plasma-wall circuit element, consisting of two nonlinear sheaths and the timevarying, nonuniform, anisotropic plasma. Capacitive sheath
currents are negligible compared to electron conduction currents. Inductively driven currents are also not significant
since the voltage and current waveforms at early times
(t,1 m s! preclude a relationship of the form V el 5LdI/dt.
Interpreting the ratio as a plasma resistance,
R pl 5V el (t)/I(t), one finds, as in Fig. 11, that the resistance
rises from a low initial value of R pl (0).2 V gradually to
R pl (1 m s).4 V at the current maximum, and then exponentially to a maximum value of R pl,max (1.3 m s).20 V at
the current disruption. Note that the plasma is not inductive
since during the disruption dI/dt,0 while V el (t).0. It is
shown in the companion paper29 that the plasma density near
the electrode during this time interval is highly depleted and
Phys. Plasmas, Vol. 4, No. 1, January 1997
current-limiting space-charge layers are created. The loss of
conductivity by plasma erosion is the principle of a plasma
opening switch.14 In those devices, inductive energy stored
in a transmission line can be rapidly transferred to a resistive
load in parallel to the plasma switch. In our experiments, the
resistance decreases during the slow recovery period of the
current (2,t,3 m s) but does not return to its original
value. Since the plasma impedance is predominantly resistive, the product V el (t)I(t)5 P pl (t) represents the instantaneous power dissipated in the plasma. Figure 12~b! shows
that the peak dissipation ( P pl,max .1250 W! occurs in the
early phase of the current disruption (t.1.2 m s!. At this
time, the energy dissipated, * P(t)dt.500 m J, in a volume
p r 2 l.10 cm2 at a density n e .531011 cm23 would increase the particle energy by 1 m J/cm3 /(831028 As/cm3 )
.12.5 eV. This value represents an upper limit of observable
heating, since much of the energization occurs in the sheath
rather than in the plasma volume. Hence some of the heat is
convected by electrons into the electrode. Some energy is
required to accelerate ions out of the current channel, while
R. L. Stenzel and J. M. Urrutia
33
FIG. 12. Effect of the external circuit properties on electrode current and
voltage waveforms for a 2 cm diam, one-sided disk electrode with n̂i B0 . ~a!
Current, I(t), open-loop voltage, V 0 5 const, and electrode voltage,
V el (t), which differs from V 0 by inductive and resistive voltage drops along
the line between source and electrode ~see the inserted circuit diagram!. ~b!
Time-dependent power delivered to the plasma, P pl 5I(t)V el (t), and plasma
resistance, R pl 5V el (t)/I(t), vs time. Resistance and power maximize near
the current disruption.
part of the energy goes into the electromagnetic fields of the
whistler mode radiated away from the electrode, thus contributing to the radiation resistance.
The inductance of the transmission line between voltage
source and electrodes has a profound effect on the I-V characteristics for time-dependent currents, created either by
switched voltages as in the present case or by timedependent plasma parameters when V 0 is held constant. For
example, Fig. 13 shows I-V characteristics of the 2 cm diam
disk electrode at various times during the rise of the current,
I(t), shown in the insert. This early phase was not discussed
in Fig. 4, which was focussed on the current disruption and
transition to steady state. In Fig. 13~a! the voltage is measured at the electrode (V el ), while in Fig. 13~b! it is the
open-loop voltage at the generator (V5V 0 ). In the latter
case, the I-V 0 curves are nearly straight lines whose slopes
increase in time. These characteristics are dominated by the
external circuit inductance, I/V 0 .t/L, which limits the current rise (V 0 @V el ). In the former case, the I-V el characteristics are also nearly straight lines, but with much higher
slopes which gradually decrease in time. These characteristics depend only on the plasma properties. Initially, the
plasma conductance is high (I/V el .1 V 21 ), but in time it
decreases due to density erosion in the current channel as
earlier shown at a single voltage @Fig. 12~b!#.
It has been demonstrated that the current depends on the
applied voltage, the external circuit, the plasma parameters,
and the electrode. However, to close the circuit, a return
electrode is required which is usually the grounded chamber
34
Phys. Plasmas, Vol. 4, No. 1, January 1997
FIG. 13. Current-voltage characteristics of a 2 cm diam, one-sided disk
electrode during the rise of the current ~see the insert!, demonstrating the
effect of the inductance of the transmission line to the electrode. When the
voltage is measured at the electrode ~a!, the slope of the curves reflects a
decreasing plasma conductance caused by plasma erosion. When the current
is plotted vs open-loop generator voltage ~b!, the curves are dominated by
the external circuit inductance, I/V 0 .t/L, i.e., form nearly straight lines
whose slope increases in time.
wall. Alternatively, floating electrodes of different sizes have
also been used. As in any passive double-probe system, the
negative electrode collects an ion current equal to the electron current collected at the positive electrode, and whichever is limited determines the maximum current. Due to the
large surface area of the present plasma column
@ 2 p r pl (r pl 1l pl ).105 cm2 , r pl .50 cm, l pl .250 cm#, the
ion saturation current to the chamber wall (I i,sat .500 A!
exceeds the peak electron currents of the small positive electrodes. However, for smaller return electrodes or smaller
plasma devices, the electron overshoot current may be limited by the return ion saturation current, and no overshoots
are observable when (I i,sat ) return ,(I e,sat ) collector or
A return /A collector , Am i /m e .271. In this case, most of the
applied potential drops off at the ion-rich sheath and the
positive electrode remains biased below the plasma potential.
Instead of a passive ion collector, the return electrode can
also be an active electron emitter, i.e., a cathode. Such
emitter-collector systems have been used for electrodynamic
tether experiments in space.4 Laboratory models have shown
the same current transients and instabilities as described
above.39
IV. CONCLUSION
Switched electron currents collected by biased electrodes
in a magnetized plasma present a challenging problem in
nonlinear plasma physics. To our knowledge, the theoretical
analysis has only been performed in the linear regime for a
two dimensional problem by solving the equations via nuR. L. Stenzel and J. M. Urrutia
merical techniques.40 Classical Langmuir probe theories are
inadequate to explain the phenomena observed. These include a transient current overshoot and a continuous, largeamplitude current instability. The present investigations have
been concentrated on external current and voltage measurements as a function of several experimental variables. While
these results provide valuable information and can lead to
plausible explanations, a conclusive physical picture arises
only in conjunction with internal measurements of fields,
currents and plasma parameters, which are presented in the
companion paper.29 Nevertheless, one can firmly conclude
from the present observations that the scaling of the transient
current overshoot with voltage and electrode size involves
the penetration of electric fields far beyond the Debye sheath,
that a density perturbation occurs on the scale length of the
electrode, and that the current-driven instability involves an
ion acoustic mode with wavelengths set by the electrode
size. Several useful conclusions for applications can also be
drawn: Diagnostic probes should be of size less than an electron Larmor radius or employ ion saturation currents. Transient electron currents well in excess of the electron saturation current can be drawn provided a sufficient return current
can be collected. The duration of the current transient is determined by the electrode dimension across B0 and the sound
speed, and the current rise time is limited by electrode voltage and circuit inductance. The current disruption caused by
the perturbation to the plasma can be used to transfer the
magnetic energy stored in the external inductance to a load
in parallel with the plasma with a voltage transformation
ratio 11(LdI/dt)/V 0 . For disk electrodes collecting crossfield currents, there is no ‘‘magnetic insulation’’ due to the
Hall effect.
ACKNOWLEDGMENTS
The authors gratefully acknowledge discussions with Dr.
Christopher L. Rousculp.
The authors also gratefully acknowledge support for this
work by the National Science Foundation under Grant No.
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