Large-amplitude oscillations in a Townsend discharge in low-current 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

diffbr0

stationary when RRR

VVj

• This circuit is commonly use for experimental and theoretical studies of oscillations and stability of Townsend discharge

• Usually, the value of the resistor R is such that the major portion of the external voltage drops across the discharge gap ( ).

• Our consideration also includes the case

• A.V.Phelps et al., 1993

• V.N. Melekhin and N.Yu. Naumov,

1985

External applied voltage V0=const

brbr0 VVV

brbr0 VVV

~

Barrier Discharge Circuit

dt

tdVCj

)(dielstationary

V

t

• Townsend discharge in barrier discharge geometry is used for addressing micro-discharge cells in plasma televisions

• 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

• Small parameter

Basic Equations Perturbation Procedurefor oscillations of small amplitude

and frequency 0 < < e-1

L

x

dx

eii jx

jx

nt

e

ecath

ineEx

4

Complementing circuit equation

dischcathcath)1(

4

1jjE

dt

di

,0anode

ij cathcath ie jj

Boundary conditions

14 disch

0

0ch

j

E

EL

ii

• Electron transit time across the gap

0oscill ee nT

• First, we derive the equation for perturbation of ion density and solve it by method of successive approximations

• Then, we immediately obtain the expression for the discharge impedance

where is proportional to perturbation of the number of ions in the gap at frequency coefficients a, b, and c depend on the parameters of steady-state discharge and the shape of ion density perturbation

Discharge Impedance

chch

2

chdisch ~

~

4

ci

aN

bNi

ZL

S

i

N~

The dispersion equation for natural oscillations in the circuit with impedance Zcirc:

0)()( circdisch ZZ

For the standard circuit

sh1circ

1)(

CiRZ

For the barrier discharge circuit

dielcirc

1)(

CiRZ

Spectrum of oscillations • In the zeroth approximation in parameter ( ) 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

0dischj

LELEi

LEi

i

i

)(

1

)(

1])(exp[

00

0

- High-frequency harmonics ( ) quickly damp in time ( ) without perturbation of the total charge in the gap

1~Im im

,...2,1,2~Re 1 mm im

- Low-frequency harmonic oscillates slowly with the frequency

Perturbation of the ion density has the same shape in the gap as the stationary ion density does

xEii enn )(stationary

01~

.10

i ~

(case )

Low frequency oscillations in standard circuit

R

i

oscillation region

ck*k *4k

02 a

0c

00 ba0diff RR

00 ckb cR

i

Stability triangle in the plane of circuit parameters and disch

sh

C

Ckc

RCi

R

i

disch

0)(diff ER

Nonlinear Low-Frequency Oscillations• Remarkable feature of the system is

that large pulses of the particle current cause small oscillations of the electric field

• Spatial distortion of the electric field due to the volume charge is quite small

21stationary

21

stationaryparticle

0

0

j

jj

E

EE

~

• Ion current in large amplitude oscillations is distributed in the gap almost the

same way as the ion current in steady-state discharge

stationary

particle

0

anodecathode

j

j

E

EE

~

xEi ej )( 01

• We consider the case when Rdiff is negative and does not depend on discharge current

0)(diff ER

• 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.

QPH QP

e1e 2

Q

HP

P

HQ

,

j

V

• Equation of separatrix in dimensionful parameters:

!0, diffdiff0 RjRVV

Hamiltonian function on the hypotenuse of the stability triangle

R

Rdiff2

jVPjQ 21ln,ln

Phase curves in general case

• Phase curve corresponding to the circuit parameters inside the stability triangle

j

V

j

V

• Phase curve corresponding to the circuit parameters outside the stability triangle

• 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.

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

• 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

0.3vacuum • Simulation parameters: Ne gas with pressure = 500 Torr, gap length = 400 m,

Vbr = 200 V, number of ions used in simulations ~ 105

11e L

Numerical Experiment• Standard circuit with R=Csh=0 (constant current source, j=8A/cm2).

The parameter = 0.02.

Voltage across the discharge gap vs. time; red lines correspond to damping rate predicted by the theory

particlej

stationaryj

Particle current vs. time; red line is analytical solution (when damping is neglected)

• Period of nonlinear oscillations

stationary

max21

~j

jT i

• Minimum value of particle current

stationarymaxe~ maxminjjjj

• In general, two types of oscillations can be distinguished in Townsend discharge:

Summary

• Short-living high-frequency oscillations If initial total charge in the gap Qinitial = Qstationary, oscillations damp in time ~ i

-1

• Long-living low-frequency oscillationsArbitrary distribution of ion density in the gap in short time ~ i

-1 takes the universal shape - the shape of ion density distribution in the steady-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.

xi en 1

• For certain parameters of the circuit the small-amplitude oscillations are self-sustained. 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

10 • Townsend discharge oscillations can be unstable (at large R) even when

0diff R