Download Large-amplitude oscillations in a Townsend

Large-amplitude oscillations in a Townsend discharge in lowcurrent limit
Vladimir Khudik, Alex Shvydky (Plasma Dynamics Corp., MI)
Abstract
We have developed a regular analytical approach to study oscillations in a
Townsend discharge when the distortion of the electric field in the discharge gap
due to the spatial charge is small. In presented theory the secondary electron
emission coefficient can take any value between zero and one. We have found that
the large-amplitude oscillations of the particle current in the discharge gap are
accompanied by small-amplitude oscillations of the gap voltage. Surprisingly, for
certain impedances of the external electrical circuit, this highly dissipative system
is governed by the Hamiltonian equations (so that the amplitude of the oscillations
slowly changes in time). Direct Monte Carlo/particle-in-cell simulations confirm
the theoretical results.
Standard Circuit
• This circuit is commonly use for
experimental and theoretical studies
of oscillations and stability of
Townsend discharge
V0  Vbr
jstationary 
when R  Rdiff
R
• Usually, the value of the resistor R
is such that the major portion of the
external voltage drops across the
discharge gap ( V0  Vbr  Vbr).
• Our consideration also includes the
case V0  Vbr  Vbr
External applied voltage V0=const
• A.V.Phelps et al., 1993
• V.N. Melekhin and N.Yu. Naumov,
1985
Barrier Discharge Circuit
• Townsend discharge in barrier
discharge geometry is used for
addressing micro-discharge cells in
plasma televisions
dV (t )
jstationary  Cdiel
dt
V
• L.F. Weber, 1998
• V.P. Nagorny et al., 2000
• To realize a dc Townsend discharge
in this circuit, external applied
voltage must change linearly with
time
t
Basic Equations
Perturbation Procedure
L



e
ni 
ji  
je
t
x
x

dx
x
e
cath
• Small parameter

E  4e ni
x
1 d
E cath  (1   ) ji
4 dt
cath
 jdisch
Boundary conditions
anode
 0,
je
cath
  ji
cath
• Electron transit time across the gap
t e  Toscill  ne  0
  t i t ch
L

 i E0
E0
 1
4 jdisch
• First, we derive the equation for
perturbation of ion density and
solve it by method of successive
approximations
Complementing circuit equation
ji
for oscillations of small amplitude
and frequency 0 < w < te-1
• Then, we immediately obtain the
expression for the discharge
impedance
Discharge Impedance
b
~
iw N w 
S
4 L
Z disch 
t ch
~
w 2 Nw 
a
t it ch
 iw
c
t ch
~
where Nw is proportional to perturbation of the number of ions in the gap at
frequency w ; coefficients a, b, and c depend on the parameters of steady-state
discharge and the shape of ion density perturbation
The dispersion equation for natural oscillations in the circuit with impedance Zcirc:
Z disch (w )  Z circ (w )  0
For the standard circuit
Z circ (w ) 
1
R 1  iw Csh
For the barrier discharge circuit
Z circ (w )  R 
1
iw Cdiel
Spectrum of oscillations
• In the zeroth approximation in parameter  ( jdisch  0 ) there exist two
different types of oscillations
• Oscillations in the system “external circuit + capacitance of empty discharge
gap”
• Oscillations of the discharge current described by the dispersion equation
exp[ iwt i   ( E0 ) L]  1
1

iwt i   ( E0 ) L
  ( E0 ) L
- High-frequency harmonics ( Re wm ~ 2mt i 1 , m  1,2,...
)
quickly damp in time ( Im wm ~ t i )1 without perturbation of the total
charge in the gap
- Low-frequency harmonic oscillates slowly with the frequency
w0   t i1.
Perturbation of the ion density has the same shape in the gap as the stationary
ion density does
 ( E0 ) x
~ n
n

1

e
i
i stationary
Low frequency oscillations in standard circuit
ti
tR
ti
  b0 k c  c0 
tR
R  Rdiff  0
a0 b0
2 a0
oscillation region
 c 0
k*
Stability triangle in the plane of circuit parameters kc 
ti
ti

t R Cdisch R
(case Rdiff   ( E )  0 )
4k*
kc
Csh
and
Cdisch
Nonlinear Low-Frequency Oscillations
• Remarkable feature of the system is
that large pulses of the particle current
cause small oscillations of the electric
field
E  E0
E0
 
jparticle  jstationary
12
• Spatial distortion of the electric field
due to the volume charge is quite small
Ecathode  Eanode
12
stationary
j
E0

jparticle
jstationary
• Ion current in large amplitude oscillations is distributed in the gap almost the
same way as the ion current in steady-state discharge
ji  1  e  ( E0 ) x
• We consider the case when Rdiff is negative and does not depend on discharge
current
Rdiff   ( E )  0
Hamiltonian function on the hypotenuse of
the stability triangle
H
e P


1 2
H 
H

Q
, P
P
Q

P  e  Q

 
2
Rdiff
R
Q  ln j, P  ln 1   V   j
• Blue line (P =  separate the regions of
stability and instability: oscillations
with relatively small amplitude are
stable and oscillations of sufficiently
large amplitude are unstable.
• Equation of separatrix in dimensionful
parameters:
V  V0  Rdiff j, Rdiff  0!
Q
2

V
j
Phase curves in general case
• Phase curve corresponding to the circuit
parameters inside the stability triangle
• Phase curve corresponding to the circuit
parameters outside the stability triangle
V
V
j
• More accurate analysis shows that in the case of circuit parameters outside the
stability region close to hypotenuse, phase curves asymptotically approach
limiting cycle (and do not “infinitely” depart from the steady state)
• Size of the limiting cycles is a sharp function of the distance to the hypotenuse.
j
Monte-Carlo/PIC simulations
• Voltage across the discharge gap is always close to the breakdown voltage so that
for not too small secondary electron emission coefficients, the spatial distribution
of ions created by the electron avalanche is different from exponential one
 eL  1  1
• Secondary electron emission coefficient depends on the properties of the cathode
surface, gas pressure, and the magnitude of the electric field. To avoid these
complications, we assumed that all electrons emitted from the cathode have zero
energy, so that always
   vacuum  0.3
• Simulation parameters: Ne gas with pressure = 500 Torr, gap length = 400 m,
Vbr = 200 V, number of ions used in simulations ~ 105
Numerical Experiment
• Standard circuit with R=, Csh=0 (constant current source, j=8A/cm2).
The parameter  = 0.02.
jparticle
jstationary
Voltage across the discharge gap vs. time; red lines
correspond to damping rate predicted by the theory
• Period of nonlinear oscillations
ti
T ~ 12

jmax
jstationary
Particle current vs. time; red line is analytical
solution (when damping is neglected)
• Minimum value of particle current
jmin ~ jmax e
 jmax jstationary
Summary
• In general, two types of oscillations can be distinguished in Townsend discharge:
• Short-living high-frequency oscillations
If initial total charge in the gap Qinitial = Qstationary, oscillations damp in time ~ ti-1
• Long-living low-frequency oscillations
Arbitrary distribution of ion density in the gap in short time ~ ti-1 takes the
universal shape n  1  e x - the shape of ion density distribution in the steadyi
state discharge. This fact allows one to eliminate the dependence of ion current
and electric field on spatial variable and obtain the equations for their amplitudes.
• For certain parameters of the circuit the small-amplitude oscillations are selfsustained. For these parameters, the system is almost Hamiltonian, so that the
amplitude of nonlinear oscillations changes in time very slowly only due to small
non-Hamiltonian terms.
• All analytical results are obtained for arbitrary secondary emission coefficient
0   1
• Townsend discharge oscillations can be unstable (at large R) even when
Rdiff  0