DTIC ADA586410: Physics Based Lumped Element Circuit Model for Nanosecond Pulsed Dielectric Barrier Discharges

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JOURNALOFAPPLIEDPHYSICS113,083301(2013)
Physics based lumped element circuit model for nanosecond pulsed
dielectric barrier discharges
ThomasUnderwood,1,a)SubrataRoy,1,b)andBryanGlaz2
1AppliedPhysicsResearchGroup,DepartmentofMechanicalandAerospaceEngineering,
UniversityofFlorida,Gainesville,Florida32611,USA
2VehicleTechnologyDirectorate,U.S.ArmyResearchLaboratory,AberdeenProvingGround,Maryland
21005,USA
(Received25December2012;accepted5February2013;publishedonline22February2013)
This work presents a physics based circuit model for calculating the total energy dissipated into
neutralspeciesfornanosecondpulseddirectcurrent(DC)dielectricbarrierdischarge(DBD)plasmas.
Based on experimental observations, it is assumed that the nanosecond pulsed DBD’s which have
been proposed for aerodynamic flow control can be approximated by two independent regions of
homogeneouselectricfield.Anequivalentcircuitmodelisdevelopedforbothhomogeneousregions
basedonacombinationofaresistor,capacitors,andazenerdiode.Insteadoffittingtheresistanceto
an experimental data set, a formula is established for approximating the resistance by modeling
plasmas as a conductor with DC voltage applied to it. Various assumptions are then applied to the
governing Boltzmann equation to approximate electrical conductivity values for weakly ionized
plasmas.Thedevelopedmodelisthenvalidatedwithexperimentaldataofthetotalpowerdissipated
byplasmas.VC 2013AmericanInstituteofPhysics.[http://dx.doi.org/10.1063/1.4792665]
I. INTRODUCTION withrelevantphysicsthatcouldbeimplementedasanapprox-
imation for the energy dissipated within a plasma for any
The need for improved control over aerodynamic flow
pulsed direct current (DC)dielectric barrier discharge (DBD)
separation has increased interest in the potential use of
configuration.Amongtheothergoalsinthispaperistoprobe
plasmaactuators.Theinherentadvantagesofplasmaactuator
into the background processes that occur within plasmas and
flowcontroldevicesinclude:fastresponsetime,surfacecom-
incorporatethatknowledgeintothemodel.
pliance,lackofmovingparts,andinexpensiveness.However,
it has been established that the actuators which affect the
flow via directed momentum transfer are not effective at
II. LUMPEDELEMENTCIRCUITMODEL
Mach numbers associated with most subsonic aircraft appli-
cations.Recently,Roupassovetal.1demonstratedthatpulsed Oneoftheprimaryassumptionsincreatingthismodelis
that nanosecond pulsed DBD’s can be approximated by two
plasma actuators, in which energy imparted to the flow
independentregionsofhomogeneouselectricfield.Onesuch
appears to effectively control flow separation, seem to be
suitable at Mach numbers (M (cid:2) 0.3) beyond the capabilities region, dubbed the “hot spot” is the region adjacent to the
powered electrode. This region makes up a small portion of
ofthecurrentplasmainducedmomentumbasedapproaches.
the total discharge area but was observed to be an important
Given the fundamental differences between the novel
pulsed discharge approach and the more conventional mo- component of the plasma discharge and necessary to obtain
agreementwithexperimentallymeasuredshockwavedynam-
mentumbasedapproaches,thereisaneedtodevelopaneffec-
tive and efficient model for the energy delivered to the flow ics by Roupassovet al.1 Theother region,dubbed the“tail,”
by the plasma. Once calculated, that value can be input to a encompasses the rest of the plasma discharge and extends to
the edge of the dielectric. As both regions are independent,
computationalfluiddynamicssolverasanenergysourceterm
themodelpresentedinthispaperconsistsofasinglenetwork
resulting in a coupled fluid/plasma dynamics model.
Multiphysicsmodelsofthistypearerequiredinordertostudy foreachregioncontainingaresistor,capacitors,andadiode.
As shown in Fig. 1, circuit elements that were used to
detailed flow characteristics. However, detailed numerical
simulations involving plasma kinetics are computationally model the plasma include: an air capacitor Ca, a dielectric
prohibitivefor a varietyof coupled fluid/plasma designprob- capacitorCd,aresistorRf,andazenerdiodeDf.Theairca-
pacitorrepresentsthecapacitancebetweenthedielectricsur-
lems. To address this issue, efficient circuit element models
face and the exposed electrode. The dielectric capacitor
have been introduced to approximate the complex processes
within plasmas. Circuit models such as those by Orlov2 rely representsthecapacitancebetweenthedielectricsurfaceand
insulatedelectrodeandisproportionaltothethicknessofthe
on empirical constants which may not be applicable to nano-
dielectriclayer.Thus,thedielectriclayerintheformofboth
second pulsed discharges. To date, an approximate model of
nanosecond pulsed plasma actuators has not been developed. its thickness and the value of its dielectric constant plays an
importantroleindeterminingtheeffectivenessoftheplasma
This paper deals primarily with establishing a flexible model
actuator. Finally, the zener diode, introduced by Orlov2 is
utilized in the model to enforce an energy threshold value
a)UndergraduateStudent.
b)Electronicmail:roy@ufl.edu.URL:http://aprg.mae.ufl.edu/roy/. belowwhichplasmawillnotform.
0021-8979/2013/113(8)/083301/11/$30.00 113,083301-1 VC 2013AmericanInstituteofPhysics
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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Physics based lumped element circuit model for nanosecond pulsed
5b. GRANT NUMBER
dielectric barrier discharges
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Mechanical and Aerospace Engineering,Gainesville,FL,32611
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083301-2 Underwood,Roy,andGlaz J.Appl.Phys.113,083301(2013)
FIG.1.Electriccircuitmodelofadielectricaerodynamicplasmaactuator.
Since a uniform charge distribution along the top of the
dielectricisassumed,thetypicalasymmetric2Dplasmaactu-
ator geometry featured in Fig. 1 can be simplified to a series
ofhomogeneoussymmetricregions.AsshowninFig.2,these
regions include: an anode sheath, “hot spot,” and “tail.” This
assumption results in a series of coupled 1D models that
accountfor the chordwisevariationalong the actuator.Fig.3
shows the simplified circuit model within each homogeneous
region.
A. Circuit FIG.3.Regionofhomogeneouspotential,i.e.,“hotspot.”
As displayed in Fig. 1, the lumped element circuit is a (cid:2) (cid:2) A
function of the two capacitance values, Ca and Cd. In this Cd ¼ 0hd d; (2)
d
model, the air is treated as both a conductor to generate a
physicalrelationshipfortheresistanceRf andaparallelplate where Ad is the cross-sectional area of the dielectric capaci-
capacitor to generate Ca. An advantage of modeling the tive element and hd is the height of the dielectric barrier
plasmaasaconductorinadditiontoaparallelplatecapacitor layer. As displayed in Fig. 5, A is the product of the span-
d
is that it generates a physical relationship for the resistance, wiselengthoftheactuatorz andd ,thewidthofthedielec-
a d
Rf, a value that is traditionally empirically determined. The tricregion.
airgapcapacitorcanbemodeledas3 Treatingtheplasmaasaconductor,theresistanceforDC
voltageisproportionaltor ,A ,andh :Itcanbegivenas3
p a a
(cid:2) (cid:2) A
Ca ¼ 0haa a; (1) Rf ¼rhAa : (3)
p a
where A is the cross-sectional area of the air and h is the
a a
approximate height of the plasma region of interest. The StartingfromKirchoff’scircuitlaws,3thegoverningdif-
heightoftheplasmahasbeenshownbyRoupassovetal.1to ferentialequationforthevoltagedropexperiencedbytheair
be approximately independent of applied voltage for nano- gap,DV,isgivenby
second pulsed DBD actuators. As displayed in Fig. 4, A is
a
the product of the spanwise length of the actuator za and la, dDVðtÞ¼(cid:3)dVapp(cid:2) Ca (cid:3)1(cid:3)(cid:3)j DVðtÞ ;
thechordwisedistancefromtheexposedelectrodetotheend
dt dt C þC R ðtÞðC þC Þ
a d f a d
ofthedielectricregion.
(4)
The capacitive element corresponding to the dielectric
canbemodeledas3 (cid:4)1 if jEj>E
j¼ crit (5)
0 if jEj(cid:4)E ;
crit
where V is the applied voltage and j is the contribution
app
fromthezenerdiode.Iftheelectricfieldmagnitude,givenas
FIG.2.Plasmadischargeregionsanalyzedinthisstudy. FIG.4.Sketchofthecapacitiveairelement.
083301-3 Underwood,Roy,andGlaz J.Appl.Phys.113,083301(2013)
thephysicalevolutioninthenumberofparticles,fðv;J;r;tÞ,
defined such that fðv;J;r;tÞ dv is the number of particles in
a unit volume located at point r, time t, internal quantum
number J, and differential velocity range vþdv. Using this
distribution function, the number of particles at point r and
FIG.5.Sketchofthedielectriccapacitiveelement. timetcanbedefinedas
jDVj ð
jEj¼ ; (6) X
h Nðr;tÞ¼ fðv;J;r;tÞdv: (9)
a
J
isgreaterthanthebreakdownelectricfield,2E ,requiredto
crit
This distribution function allows a mathematical
ionizeair,thenjtakesonavalueofone,otherwiseitiszero
descriptiontobedevelopedforthetemporalevolutioninthe
tosignifythatplasmahasnotformed.
number of particles resulting from particle collisions within
a control volume. The time rate of change in the number of
B. Conductivity
particlesduetoexternallyappliedfieldscanbedescribedas5
To effectively calculate the resistance governed by Eq.
Df fðvþdv;J;rþdr;tþdtÞ(cid:3)fðv;J;r;tÞ
(3), an expression must first be developed for the electrical ¼ : (10)
conductivity, r , of the plasma. This value is one that tradi- Dt dt
p
tionallyrequiresanumericalapproach.Tosimplifytheprob-
lem to a point where an analytic formulation can be used, A partial differential equation can be developed to
numerous simplifying assumptions were used and are describeEq.(10)usingformulasforthetimerateofchangeof
describedinthefollowingparagraphs. vandrestablishedusingtheequationofmotionintheformof
For any plasma, the resulting electric current is com-
posed of two primary terms: the current from electrons and dv F
¼ ; (11)
that from ions. As the drift velocity of electrons, w , in a dt m
e
non-equilibrium plasma is significantly higher than ions, the
dr
current density can be approximated as the portion from ¼v; (12)
electrons if the number densities, N and N, are approxi- dt
e i
mately the same. Using a form of the generalized Ohm’s
and the chain rule of calculus to obtain the Boltzmann ki-
Law, the current density vector, J and r , respectively, can
p neticenergyequationgivenas
bewrittenas
J(cid:2)(cid:3)eN w ¼r E; (7) Df @f @f F @f
e e p ¼ þv(cid:5) þ (cid:5) : (13)
Dt @t @r m @v
r ¼eðN l þNlÞ; (8)
p e e i i In order to approximate the total derivative, a relaxation
time can be introduced defined as the time taken for the sys-
where l and l represent the ion and electron mobilities,
i e temtobereducedtoanequilibriumdistributionfunction.The
respectively. Much like Eq. (7), the electrical conductivity
tau approximation can be given as s(cid:2)ðN r jvjÞ(cid:3)1, where
relation can be simplified using the concept of quasineutral- a ea
r isthecollisioncrosssectionbetweenelectronsandatoms,
ity which is defined as having approximate equal number ea
v istheaveragecollisionvelocity,andN isthenumberden-
densities for charged particles of opposite polarity. Thus, as a
sityofatoms.5Ifthenumberofparticleswithinacontrolvol-
l istypicallythreeordersofmagnitudelargerthanl,itisa
e i umeisdefinedasaequilibriumdistributionfunction,f ,when
good assumption to approximate the electrical conductivity 0
each particleis at the same energy level as its neighbor, then
asonlycomingfromelectronsaslongasN isatleastofthe
e the particle evolution over timedue to pairwise collisions af-
same order of magnitude as N.4 Quasineutrality is a typical
i teranexternalforcehasbeenappliedcanbegivenas5
assumption that is valid as long as the plasma being model-
ing is far enough away from the powered electrode to avoid
Df f (cid:3)f
theboundarylayerinplasmaphysicscalledthesheath. ¼(cid:3) 0: (14)
Dt s
Since a pulsed DC voltage is assumed, the activation of
the external electric field will follow the voltage waveform As s only accounts for collisions between electrons and
asastepfunction.Thus,twoexpressionswillberequiredfor neutral atoms,itisonlyaccurate intheevent ofweaklyion-
r , where the first is valid for the period when an external izedplasma.Plasmaactuatorsconsideredinthispapertradi-
p
electric field is applied, as shown in Fig. 9 from 0 to 60ns, tionally feature a low degree of ionization, or simply the
and the second when the voltage drop over the air gap is amount of air that is ionized, and thus can be treated in a
zero. For the portions of the voltage waveform that DV is weakly ionized limit. In terms of the momentum-transfer
zero, the power is also zero according to Ohm’s Law and collision frequency which can be defined as the mass cor-
thustheconductivityduringthistimeisofnoimportance. rected rate at which a particle of a specific species collides
Togenerateaanalyticformulafortheelectricalconduc- withanother,thecriteriaforaweaklyionizedplasmacanbe
tivity, a distribution function must be introduced to describe givenas4
083301-4 Underwood,Roy,andGlaz J.Appl.Phys.113,083301(2013)
(cid:3)(cid:2) (cid:6)(cid:3)(cid:2) : (15) neglected. The electrical conductivity and electron mobility,
ei en
respectively,canbegivenas
This equation requires that the collision frequency
betweenelectronsandionsbemuchlessthanthosebetween N ðt(cid:8)Þe2ðt(cid:8) (cid:2) t(cid:3)t(cid:8) (cid:3)
r ðt(cid:8)Þ¼ e EðtÞexp dt; (19)
electrons and neutrals. Thus, if this requirement is met, the p m Eðt(cid:8)Þ N(cid:8)r(cid:8) (cid:3)(cid:8)
e 0 a ea
collisional occurrences between electrons and charged par-
ticles can be effectively ignored and the relaxation time e ðt(cid:8) (cid:2) t(cid:3)t(cid:8) (cid:3)
established is a good approximation of the total time rate of leðt(cid:8)Þ(cid:2)m Eðt(cid:8)Þ EðtÞexp N(cid:8)r(cid:8) (cid:3)(cid:8) dt; (20)
changeinthenumberofparticleswithinacontrolvolume. e 0 a ea
After introducing a relaxation time to approximate Eq. where r is a function of electron energy and can be
ea
(13), an equation of motion describing the average velocity obtained for various molecules found in air from Phelps.6
of electrons can be established by multiplying by mev and Numerical values used in the model are included in the
integrating over the electron velocity. An analytic formula- Appendix.Theelectronvelocitycanbeobtainedbyassuming
tion for the average velocity an electron experiences due to a Maxwellian velocity profile. As a collection of electrons
an externally applied field, we, can be obtained by assuming withinplasmahavearangeofvelocities,theMaxwellianve-
thattheforcetermcanbeapproximatedastheLorentzforce locity profile represents the most probable distribution of
e these velocities. Thus, the distribution of velocities,
F¼(cid:3)eE(cid:3) ðv(cid:7)BÞ; (16) fðu;v;wÞ,canbegivenby4
c
2 1 3
where B represents the magnetic field and E represents the mðu2þv2þw2Þ
electric field. As plasma actuators have no applied magnetic fðu;v;wÞ¼A3exp64(cid:3)2 k T 75; (21)
field, B can be set equal to zero. Therefore, the equation of b e
motionforanelectrondescribingw canbegivenas
e
(cid:2) (cid:3)1
m 2
A ¼N : (22)
medweðtÞþmeweðtÞ¼(cid:3)eEðtÞ: (17) 3 e 2pkbTe
dt s
Using the non-relativistic definition of kinetic energy, a
SolvingEq.(17),alinearfirst-orderdifferentialequation,an relationship can be established for the kinetic energy of an
integralequationisobtained electronthatisvalidinthelimitjvj(cid:6)c,wherecisthespeed
of light. Using this approximation, the relativistic formula-
e (cid:2) t(cid:8)(cid:3)ðt(cid:8) (cid:2) t(cid:8)(cid:3) tionofthekineticenergycanbeapproximatedas
w ðt(cid:8)Þ¼(cid:3) exp (cid:3) EðtÞexp (cid:3) dt: (18)
e m s(cid:8) s(cid:8)
e 0 mc2 1
E ¼ (cid:3)mc2 (cid:2) mv2: (23)
k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Equation (18) can be solved in conjunction with Eq. (7) 1(cid:3)jvj2=c2 2
toobtainanexpressionforthetimevaryingconductivityand
in conjunction with Eq. (8) to obtain an approximation for AveragingEq.(21)andusingthenon-relativisticdefinitionof
the time varying electron mobility if the ion mobility is kineticenergy,themeankineticenergyofelectronsbecomes4
2 1 3
ððð1 A 1m ðu2þv2þw2Þexp6(cid:3)2meðu2þv2þw2Þ7 du dv dw
32 e 4 k T 5
(cid:3)1 b e
E ¼ ; (24)
av 2 1 3
ððð1 meðu2þv2þw2Þ
A exp6(cid:3)2 7 du dv dw
3 4 k T 5
(cid:3)1 b e
where k isBoltzmann’s constant. From the definition of ki- TherequiredinputsforEqs.(19)and(20)include:N ,the
b e
neticenergy,therelationshipbetweenE andjvjwithvector numberdensityofelectrons,N ,thenumberdensityofatoms,
av a
componentsðu;v;wÞcanbeestablishedandtheaveragether- andT ,thetemperatureoftheelectrons.Amongthesevalues,
e
malvelocityofelectronsbecomes N canbe assumedtobe constantintimeasthe numberden-
a
sityofatomsissignificantlyhigherthanthatoffreeelectrons.
sffiffiffiffiffiffiffiffiffiffiffi Many other models incorporate a constant electron
3k T
jvj¼ b e: (25) temperature into their model.2,7 Using experimentally
me measured values of reduced electric field,8 E=N vs.
a
083301-5 Underwood,Roy,andGlaz J.Appl.Phys.113,083301(2013)
electrontemperatureasdetailedinFig.6,thismodelcalcu- where A and B are empirical constants that have tabulated
lates a new electron temperature at each time step by com- values of 15 and 365, respectively, for air at atmospheric
paring the E=N value experienced by the plasma, produced pressure.9
from using Eqs. (4)–(6), with Fig. 6. It is assumed in this
paperthattheeffectsofanappliedelectricfieldhaveanin-
stantaneous,on a time scale much faster than10(cid:3)9 s,effect C. Dischargedevelopment
onelectrons. Thefinalremainingunknownrequiredtoclosetheequa-
Atime-varyingdifferentialequationthatgovernsNe can tion system is to determine how the electric potential
be obtained from the drift-diffusion equations. The electron changes over the horizontal length of the actuator. To accu-
continuityequationthatgovernsNecanbegivenby rately model this, it is important to incorporate the wall
effectsoftheplasmaactuator.Intermsofpotentialvariation,
@Neþr(cid:5)C ¼ajC j(cid:3)bnn ; (26) these wall effects attract charged particles ofoppositepolar-
@t e e i e ity and shield charged particles of the same polarity.
Therefore, for regions beside the anode (powered electrode
where Ce is called the charged species flux, a is the for positive pulses), an anode sheath is developed where an
Townsend coefficient of ionization, and b is the recombina- attractionofelectronsoccursandarepulsionofpositiveions
tioncoefficient betweenelectron andneutral atoms.Bysim- occur. For regions above the dielectric region on top of the
plifying the plasma discharge into a combination of two grounded electrode, a cathode sheath is developed where
homogeneous regions plus an anode sheath and by invoking positive ions are collected and electrons are repelled. Fig. 7
the irrotational property of electric fields, Eq. (26) can be illustratesthetwopredominate sheathsthataredevelopedin
simplified by ignoring any spatial variation in the number asymmetricplasmaactuators.
densityofelectrons,i.e.,Ce becomes Itisimportanttoconsidertheeffectsofchargecollection
andrepulsionintheseregionsassuchphenomenoncanhave
Ce (cid:2)NelejEj: (27) largeeffectsonthevariationoftheelectricpotentialoverthe
chordwiselengthandheightoftheplasmadischarge.
Assuming the number densities of electrons and ions are
equal, i.e., in the quasineutralregion,the electron continuity
equationforair,withacompositionof80%N and20%O , 1. Walleffects
2 2
canbewrittenas
As the model introduced in this paper is interested in
solvingfortheamountofenergytheplasmatransferstoneu-
dN
e (cid:2)a jN l Ej(cid:3)0:80b N2(cid:3)0:2b N2; (28) tral species, the relative importance of both the anode and
dt air e e N2 e 02 e
cathodesheathregionsneedstobeestablished.Energy,Q,is
a function of electrical conductivity and electric field
where a and b, respectively, can be approximated as hav-
air
ingtheform9 strengthandcanbegivenas
(cid:2) (cid:3)
Bp Q¼r jE2jV : (32)
a ¼Apexp (cid:3) ; (29) p vol
air
E
Equation (32) shows that the energy transfer from both
(cid:2) (cid:3)
300
b ¼2:8(cid:7)10(cid:3)7 ; (30) sheath regions is proportional to the conductivity of the
N2 Te plasmawithineachregion.Asthecathodesheathregiontyp-
ically has very few electrons due to repulsion effects, the
(cid:2) (cid:3)
300
b ¼2(cid:7)10(cid:3)7 ; (31) current within this region will be carried by positively
02 Te chargedions.Themobilityofsuchionsaresignificantlyless
than that of an electron, so as a first order approximation if
the local charged species number density and electric field
strength are assumed equal, the conductivity in the cathode
sheath will be significantly less than the conductivity in a
bulkplasmaandtheanodesheath,i.e.,
FIG.7.Collectivewalleffectsoftheexposedpoweredelectrodeandvirtual
FIG.6.Plotofelectrontemperaturevs.reducedelectricfield. electrode(dielectricregion).(1)Anodesheathand(2)cathodesheath.12
083301-6 Underwood,Roy,andGlaz J.Appl.Phys.113,083301(2013)
r (cid:6)r (cid:2)r : (33) equal number densities of both electrons and ions (n ¼n),
s(cid:3)c p s(cid:3)a e i
while the cathode and anode sheaths only have electron and
When combined with the fact that the volume of both ion densities, respectively. If this is assumed then the anode
sheath regions are orders of magnitude less than the bulk sheath can be approximated as having a net charge density
plasma, an order of magnitude approximation to the plasma equal to the quasineutral region’s calculated electron number
discharge can be obtained by ignoring the cathode sheath’s density.Byassumingthe anode sheathisdevoidof anyposi-
effects for nanosecond pulsed plasma discharges. Although tive ions and has a time-dependent number density of elec-
the cathode sheath is approximately negligible in terms of trons,Poisson’sequationforthesheathcanbegivenby
energy transfer, the higher electric field strength and higher
conductivity present in the anode sheath contain important d2/ eN ðtÞ
physics necessary for capturing the energy transferred by a dx2 ¼ (cid:2)e : (41)
0
plasma discharge. Fig. 2 shows the updated “hot spot” and
“tail” regions with the anode sheath included in the hot spot If this form is assumed for a single Debye length (k ), then
D
regionadjacenttothepoweredelectrode. theremainingdischarge(restof“hotspot”and“tail”)region
is part of the quasineutral bulk plasma and can be given by
2. Electricpotentialvariation Laplace’sequation
Toestablishavariationintheelectricpotential,thegov-
erningMaxwellequationscanbeusedwhicharewrittenas3 d2/
¼0: (42)
dx2
q
r(cid:5)E¼ f; (34)
(cid:2) Theordinarydifferentialequationsforelectricpotentialvari-
0
ation in the hot spot and tail regions, respectively, can be
r(cid:5)B¼0; (35) solvedtoprovideanapproximate1Dspatialvariation
@B
r(cid:7)E¼(cid:3) ; (36) 8eN
@t < ex2þC xþC if x(cid:4)k
/ðxÞ¼ 2(cid:2)0 1 2 D (43)
@E :
r(cid:7)B¼l Jþl (cid:2) ; (37) C3xþC4 if x>kD:
0 0 0 @t
Equation (43) requires a total of 4 boundary conditions.
in differential form where q ¼eðn (cid:3)n Þ is the net charge
f i e Thosecanbesummarizedas
density and l is the permeability of free space. In the ab-
0
sence of a time-varying magnetic field, Eq. (36), Faraday’s
/ ðx¼0Þ¼V ; (44)
1 app
LawofInduction,simplifiesto
/ ðx¼LÞ¼V ; (45)
r(cid:7)E¼0; (38) 2 break
/ ðx¼k Þ¼/ ðx¼k Þ; (46)
1 D 2 D
andsincethecurlofEiszero,theelectricfieldcanbesolved
for as a potential function / and substituted into Gauss’ d/ d/
1ðx¼k Þ¼ 2ðx¼k Þ; (47)
Law, Eq. (34). The resulting equations, respectively, can be D D
dx dx
givenby
where/ isthepotentialintheanodesheath,/ isthepoten-
E¼(cid:3)r/; (39) 1 2
tial inthe quasineutral region, L islength of the actuator, and
q k is a Debye length. The first boundary condition is V at
r2/¼(cid:3) f: (40) D app
(cid:2) thecathodeandthesecondisbasedontheexperimentalobser-
vationsbyRoupassovetal.1Itwasobservedthataplasmadis-
Thenetchargedensitywithinthequasineutralregionofa charge could be approximated as stopping at the edge of the
plasma is equal to zero as n ¼n. For the anode sheath, the grounded electrode for asymmetric actuators independent of
e i
net charge density can be approximated if the plasma is the applied voltage; so, the edge represents the absolute limit
assumedtouniformlydistributeitschargedensity.Therefore, of ionization or the breakdown voltage of air. Using Eqs.
the quasineutral region of the “hot spot” and “tail” features (44)–(47)asboundaryconditions,thepotentialbecomes
8 (cid:6) (cid:7)
eN eN eN V (cid:3)V
>>><2(cid:2)ex2þ 2(cid:2) Le k2D(cid:3) (cid:2) ekDþ breakL app xþVapp if x(cid:4)kD
0 0 0
/ðxÞ¼ (48)
(cid:6) (cid:7)
>>>: 2e(cid:2)NeLk2DþVbreakL(cid:3)Vapp ðx(cid:3)LÞþVbreak if x>kD:
0
083301-7 Underwood,Roy,andGlaz J.Appl.Phys.113,083301(2013)
The Debye length provides an order of magnitude TABLEI.Experimentalparameters.1
approximationfortheextentofaplasmasheathbyassuming
an exponential Boltzmann distribution in the charge density ha 0.4mm
within the plasma discharge. Substituting this into Poisson’s hd 0.3mm
(cid:2) 2.7
equation, d
(cid:2) 1
a
d2/ (cid:6) (cid:2) e/ (cid:3) (cid:7) An 30mm2
(cid:2)0 dx2 ¼eN1 exp k T (cid:3)1 ; (49) Ad 30mm2
b e V 50kV
DT 60ns
where N is the charged particle density far away from the
1
electrode. Taking a first-order Taylor expansion, the Debye
lengthcanbegivenas4
k (cid:9)(cid:2)(cid:2)0kbTe(cid:3)12: (50) whichallowsanumericalODEsolver,suchasthoseemploy-
D N e2 ingtheDormand-Princemethodtominimizefunctionalerror
1 by adjusting the step size after each time step.10 Numerical
integration required for Eqs. (19) and (20) and energy
Although at high voltages, a first-order approximation fails,
derivedfromthecircuitmodel,givenas
Eq. (50) still provides an order of magnitude approximation
oftheextentoftheanodesheath.
ðt DV2
Q¼ dt; (60)
R ðtÞ
D. Numericalprocedure 0 f
Whensolvingfortheenergyimpartedtoneutralspecies, were performed viathe Gauss-Kronrod quadraturemethod10
a coupled equation system results from Eqs. (4) and (28). ateachtimestep.
Thecoupledtermsinclude
III. RESULTS
R /N ; (51)
f e
A. Validationagainstexperiment
C /DV; (52)
e
Tovalidatetheaccuracyofthe modeldescribedinthis
b/DV; (53) paper, comparisons with data presented in Roupassov
etal.1areprovided.Theexperimentalparametersthatwere
/1 /Ne: (54) mentioned and used in the circuit model are given in Table
I. Reference 1 uses the electrode configuration detailed in
To solve the resulting equation system, the Dormand-Prince Fig.8.
Runge-Kutta method was employed. This method provides Fig.9illustratesanapproximationoftheappliedvoltage
an efficient way to incorporate an adaptive step size that is squarewavethatwasintroducedintheexperimentalworkof
important for computational efficiency in a problem that Roupassov et al.1 The slope that is introduced is to simulate
requires small time steps for convergence. The benefit of afunctionthatisdifferentiable.ThisisneededforV0 ðtÞin
app
such a procedure can be illustrated through a simplistic the governing differential equation as detailed by Eqs. (4)
example.Iftheerrorofeachtimestepisdefinedas and(5).Onecouldalsogenerateacontinuousfunctionusing
Fourier decomposition of a traditional square wave; how-
(cid:2)i ¼yðtÞ(cid:3)yi; (55) ever,thiswouldnotaccountfortheminorriseandfalltimes
k k
foundinexperimentation.
then if two step sizes are considered, h and h , the error of
1 2
each iteration and their relative error, respectively, can be
givenas
yðtÞ(cid:3)y1 ¼(cid:2)1 ¼ah ; (56)
n1 n1 1
yðtÞ(cid:3)y2 ¼(cid:2)2 ¼ah ; (57)
n2 n2 2
y2 (cid:3)y1 ¼aðh (cid:3)h Þ: (58)
n2 n1 1 2
Therefore, for a given error tolerance, (cid:2), a sequence of step
sizescanbegenerated
ðh (cid:3)h Þ(cid:2) FIG. 8.Experimental scheme used by Roupassov et al.1 for the discharge
hiþ2 ¼qjyiiþ1(cid:3)iþy1 ij; (59) gap. (1) High-voltage electrode; (2) dielectric layer; (3) low-voltage elec-
niþ1 ni trode,(4)zoneofdischargepropagation,and(5)insulatingplane.
083301-8 Underwood,Roy,andGlaz J.Appl.Phys.113,083301(2013)
FIG.9.Plotoftheinputvoltage,V vs.timeusedinthemodel.
app
FIG.10.Plotofelectronnumberdensityvs.timeforthe“hotspot.”
B. “Hotspot”results
finer the relative error tolerances are required to be to guar-
For the small region of 0.4mm(cid:7)0.4mm over all time
anteeconvergenceofasolution.
outsideoftheanodesheath,thetimevariationinthenumber
Using the results displayed in Figs. 10 and 11, the total
density can be established. As an initial condition for Eq.
power dissipated toneutral species as afunctionoftime can
(28), 1015 m(cid:3)3 electrons were assumed based on work by
be established. Using the result of Eq. (4) and the relation-
Ref. 1. As displayed in Fig. 10,there is a large gradient that
ship for the instantaneous power, the time-varying power
occurs during the rise time that peaks around 1:13(cid:7)1019
impartedtotheflowcanbegivenas7
m(cid:3)3. It is also evident in Fig. 10 that the recombination of
electrons is largely negligible on the nanosecond time scale. DV2ðtÞ
If a frequency of 1 kHz is used, the recombination of elec- PðtÞ¼ : (61)
R ðtÞ
f
trons with atoms allows a steady-state electron number den-
sity to be achieved on the nanosecond time scale. The
As shown in Fig. 12, the instantaneous power is domi-
recombination of electrons becomes a significant quantity
nateduringtherisetimeforthe“hotspot”region.Uponinte-
when exploring the dynamics of a plasma discharge on the
grating the instantaneous power over time using Eq. (60),
microsecondtimescale.
this model produces an energy value of 2.1 mJ for this
UsingEq.(4)andFig.6,themodelwasabletoproduce
region. When compared to the experimentally determined
atime-varyingelectrontemperature.AsdisplayedinFig.11, valueof4.2mJbyRoupassovetal.,1thismodelproducesan
there is an initial spike in the electron temperature to 29eV
order of magnitude estimate for the energy imparted to neu-
((cid:2)340000K) that coincides with the peak in electron num-
tralspeciesinthisregion.
berdensityat11ns.Fig.11alsosuggeststhattheassumption
ofaconstantelectrontemperatureisnotanaccurateassump-
C. “Hotspot”sheathresults
tion for nanosecond pulsed DBD plasmas. The variability in
theelectrontemperaturebeyond40nsisduetothenumerical Using the Debye length approximation provided by Eq.
errortolerancesthatareselectedwhensolvingEqs.(4),(19), (50)andthesolutionfromthequasineutral“hotspot”region
and(28).Fig.11showsthatwhenreducingtherelativeerror ðn (cid:9)1019 m(cid:3)3, T (cid:9)10 eV), a Debye length of 7.43 lm is
e e
intheDormand-Princemethodfrom10(cid:3)2 to10(cid:3)3,thestrong generated. The electron number density experienced in the
functional variability experiences a significant reduction. anodesheathisassumedtobeequaltothevaluescalculated
Thehigherthegradientsareduringtheriseandfalltime,the in the adjacent “hot spot” region. Fig. 13 shows the time
FIG.11.Plotofelectrontemperaturevs.
time for the “hot spot”: (a) 10(cid:3)2 error
and(b)10(cid:3)3error.
083301-9 Underwood,Roy,andGlaz J.Appl.Phys.113,083301(2013)
FIG.12.Plotofpowervs.timeforthe“hotspot.” FIG.14.Plotofelectrontemperaturevs.timeforthe“hotspot”sheath.
variation in the electron number density within the sheath
andalsoshowsapeaknumberdensityof1:13(cid:7)1019m(cid:3)3.
Using the revised form of the electric potential within
theanodesheathgivenbyEq.(48)andusingFig.6,theelec-
tric temperature variation over the duration of the pulse can
be established. As displayed in Fig. 14, there is an initial
spike in the electron temperature to ((cid:2)348000K) that coin-
cideswiththepeakinelectronnumberdensityat11nsmuch
likethehotspotregion.
AsshowninFig.15,theinstantaneouspowerisdominant
during the rise time for the anode sheath region. Upon inte-
gratingtheinstantaneouspowerovertimeusingEq.(60),this
model produces an energy value of 0.045 mJ for this region.
This number is quite small compared to the 2.1 mJ experi-
FIG.15.Plotofpowervs.timeforthe“hotspot”sheath.
enced in the “hot spot” region. However, the anode sheath
doeshaveaslightlyhigherlinearenergydensitythanthe“hot D. “Tail”results
spot” region(6J/m vs. 5.25J/m). The reason that thereisnot
For the region 0.4mm(cid:7)4.6mm, the time variation in
significantdeviation predicted inthe anodesheath andquasi-
the number density can be established. As an initial condi-
neutral“hotspot”regionsisthatexplicitchargebuildupisnot
tionforEq.(28),1015 m(cid:3)3electronswereassumed,thesame
accounted for in the circuit model within the sheath region.
number of electrons assumed for the “hot spot” region.
AsshownbyRef.11,theelectricfieldinthesheathandadja-
WhencomparingFigs.10and16,thetailregionexperiences
centquasineutralregionsareapproximatelyequaluntilcharge
a lower growth rate in the number of electrons which is due
buildup is allowed within the anode sheath during the length
to the lower electric field experienced by this region. As
ofthepulseoraseriesofpulses.
describedbyEq.(28),alowerelectricfieldproducesalower
FIG.13.Plotofelectronnumberdensityvs.timeforthe“hotspot”sheath. FIG.16.Plotofelectronnumberdensityvs.timeforthe“tail.”