Electric probe measurements of plasma oscillations in the khz range within the discharge of the PPSX000 Hall thruster

Transcription

1 Electric probe measurements of plasma oscillations in the 15 khz range within the discharge of the PPSX Hall thruster IEPC Presented at the 31 st International Electric Propulsion Conference University of Michigan Ann Arbor, Michigan USA September 2-24, 29 Jacek Kurzyna Institute of Plasma Physics and Laser Microfusion, 23 Hery Str, Warsaw, Poland and Stephane Mazoure, and Vladimir Kulaev ICARE CNRS, 1C Avenue de la Recherche Scientique, 4571 Orléans Cedex 2, France Abstract: The cross-correlation between local plasma oscillations and discharge current are studied in a high voltage (HV) PPSXML Hall eect thruster (HET). A set of Langmuir probes is used to collect plasma bulk signals in MF-range referred to as transit time oscillations. Two probe congurations are investigated: a) two probes located downstream of the thruster exit and b) three probes immersed in a slit of the outer isolator of the thruster. The main goal of the current measurements was to check whether the oscillations as captured by the probes positioned in various locations have essentially a volumetric character and are thus related to a standing wave or if they can be related to a traveling wave propagating along the thruster z axis. Both the empirical mode decomposition (EMD) method and classic correlation analysis have been used to assess the signal coherence. For dierent probe locations cross-correlation of appropriate residua of the intrinsic mode functions (or even raw signals) shows the existence of a time delay when the thruster is operated at the highest voltages. However, the phase-analysis applied to MF modes does not display the phase shift of synchronized modes that can conrm the longitudinal propagation of the corresponding wave. I. Introduction Snecma Moteurs PPSX 5 kw Hall eect thruster (HET) is a high voltage successor of PPS1 series. With discharge voltage up to 1 kv and power of 6 kw PPSX HET results in maximal thrust of 33 mn and specic impulse of 32 s. It was designed to satisfy specic needs of future space technology that requires a propulsion device capable to control the orbits of Earth satellites and ecient enough to serve as a main engine useful for deep space missions. Studies of the operational envelope of PPSX thruster indicated that it can operate eciently in two desired regimes see Ref. 1. Being under examination at ICARECNRS PPSX-LM HET is a laboratory modication of the original version of this thruster. The current studies of its plasma dynamics are the continuation of our previous ones 2 that concerned stability of the discharge and plasma oscillations induced in a wide frequency band Research Scientist at IPPLM, jkurzyna@ifpilm.waw.pl. Research Scientist at CNRS, EP team of the CARE institut, mazoure@cnrs-orleans.fr. Graduated student, EP team of the CARE institute, kulaev@cnrs-orleans.fr. 1 September 2-24, 29

2 when operating parameters of the thruster were varied. However, in the current paper we are particularly interested in the oscillations of medium frequency (MF) range that belongs to the band of 1 5 khz. In HET literature these oscillations are called transit-time as their characteristic period is scaled with the time of ion transitions at the length of thruster acceleration zone. Transit-time oscillations have been already experimentally identied in the early studies of HET discharge stability. 3 Their characteristic frequency band spreads between the band of LF oscillations that are known as a breathing-mode representing ionization instability at the length of HET channel, and the band of HF oscillations ( 5 5 MHz) that represents electrostatic wave drifting azimuthally. 47 For the more complete list of dierent type oscillations induced in HET plasma and their categorization scheme that bases on characteristic time scales and physical processes involved the reader is referred to the review paper of Zhurin Ref. 8 and Choueri Ref. 9 Despite good insight into the nature of the breathing-mode, only recently it was shown that coupling of plasma with an electric circuit should be included in the accurate modeling of these basic for HET discharge dynamics oscillations. 1 Nevertheless, the main features of the breathing-mode are well reproduced by almost any numerical model independently if it is based on uid, kinetic or hybrid formulation. Moreover, in such models usually are also reproduced the oscillations of MF band that most likely are triggered by transit-time instability. For instance, in quasi-neutral one-dimensional time dependent uid formulation 11 numerical calculations result in MF oscillations (of several hundred khz) superimposed on the LF breathing mode. These MF oscillations aect all the main physical quantities like electron and ion density n, electron temperature T e, ion and electron axial velocities v i, v e and electron azimuthal velocity v ϑ. However, discharge current I D in this model represents mostly the breathing mode and is only slightly modulated by MF component. Similarly, the hybrid model with kinetic description of ion component 12 reproduces ionization as well as transit-time oscillations. In this model transit-time oscillations appear as a periodic ( 1-4 khz) modulation of ion current and out of nominal operating conditions may strongly aect total discharge current. According to the authors they are induced due to the interaction of ions with the acceleration zone that oscillates along the thruster axis. Thus an ion beam is split into groups of fast and slow components that results in broadening of ion energy distribution function (IEDF). Consequently the eciency of the thruster decreases. On the other hand transit-time oscillations suppressing may be gained by the growth of electron current (in this model increasing e.g. anomalous electron transport towards the anode). However, the growth of electron current results in farther degradation of thruster eciency. Transit-time oscillations were also analyzed theoretically studying short-wave asymptotics of WKB approximation applied to 1D uid model of HET discharge in which cold ions are accelerated between two electrodes within quasi-neutral plasma see Ref. 13. Considering a supersonic ion ow the authors explain triggering of self-sustained transit time oscillations as a resonance interplay of longitudinal acoustic waves and discharge current perturbation ensured by a feedback mechanism in plasma volume limited by two electrodes. It is also indicated that frequency of transit-time oscillations scales inversely proportional to transit time of sonic wave in the region between the electrodes. Moreover the authors explain why experimentally studying dynamics of plasma density one can expect traveling wave while plasma potential oscillations should be pronounced rather as a standing wave. Additionally it is suggested that the oscillations may be mostly sustained by a direct transfer of energy from the ion ow to the wave. That is why transit-time oscillations may reduce mean ion velocity. Optimizing HET one is usually searching for such operating conditions in which the major part of electric power is transferred into energy of ions exhausted by the thruster. Thus, the studies of the operational envelope of HET usually results in nding of such a range of parameters in which the amplitudes of total discharge current oscillations are small when compared with its mean value. It means that LF breathingmode is minimal in these operating conditions. However, operating PPSX HET in high voltage regime we have observed 2 deep modulation of discharge current with frequency of about 5 times faster than that of conventional breathing-mode. Applying empirical mode decomposition (EMD) method 14 this MF component was ltered out from a slow mode of the amplitude several times lower than that of MF wave. Keeping in 12, 13 mind this nding and that transit-time oscillations can consume signicant part of energy of the ion beam (what leads to the degradation of thruster eciency) it is obvious that this type of oscillations should be studied more systematically in experiments. Therefore, in the current studies we are mainly interested in the categorization of the observed MF oscillations as a standing or traveling wave. Hence, the correlations of signals as captured in dierent locations along the PPSX thruster axis are examined. To make the experiment as simple as possible a set of single electrostatic probes biased with constant voltage was used. 2 September 2-24, 29

3 Our experiment is somehow similar to that of Chesta at al see Ref. 15. It was indicated in our previous papers 2, 6 that signals collected during operation of Hall eect thruster are usually non-stationary in the time scale of measurements and often of intermittent nature. That is why also in the current studies we use the EMD method for signal analysis. However, in addition to its original formulation 14 and later modication applicable for treatment of intermittencies, 16 we also applied its new formulation basing on noise driven decomposition. 17 This method is briey described in the next section. In the subsequent section the outline of the experiment is depicted. Finally, the obtained results are discussed and conclusion is drawn. II. Ensemble Empirical Mode Decomposition It was mentioned above that signals as measured in our experiment are non-stationary. Thus instead of Fourier analysis we apply (as before) self-adaptive decomposition of the signals onto a set of simpler components - intrinsic mode functions -imf's. For the description of EMD method the reader is referred to the original paper of N. Hunag Ref. 14, and our previous studies on HET plasma uctuations. 2, 6 Here just minimal information will be given to make the farther discussion readable. When a non-stationary signal is to be analyzed one can ask about its intensity and rate of variations at any moment in time. In this way instantaneous frequency and instantaneous power corresponding to it are often examined, like for frequency modulated signals. Instantaneous frequency is usually dened as a time derivative of the phase of a complex function z(t) that is to be built using a given real signal s(t) and a function s(t) that is orthogonal to this real signal s(t) (see e.g. Ref. 18). When s(t) is dened by the Hilbert transform of s(t): H[s](t) def = 1 π Pv s(τ) den dτ = s(t), (1) t τ where Pv denotes principal value of the integral, then z(t) def = s(t) + ıh[s](t) den = A[s(t)] is called an anlytical signal. Writing z(t) in the form A(t) exp(iϕ(t)), the instantaneous frequency ω I (t) of the analytical signal is dened as a derivative of the complex phase, ω I (t) def = dϕ(t)/dt, and the instantaneous power as a square of its amplitude A(t), where as usual: A(t) def = s 2 (t) + s 2 (t). However, for a wide class of signals the phase ϕ(t) can be nonmonotonic function of t. In such a case instantaneous frequency may become negative and lose its physical meaning. The signals of this type are often called multi-component. The solution to this problem consists in representing a given multi-component signal as a sum of k simpler components s k (t) (mono-component signals) with well behaved instantaneous frequency ω k (t). Thus, one can write: s(t) = N s k (t) = R [ N a k (t)exp(iϕ k (t)) ] = R[A(t) exp(iϕ(t))], (2) k=1 k=1 where each elementary analytical signal A k [s k (t)] = s k (t) + ih[s k ](t) = z k (t) is determined by s k (t) and its Hilbert transform. For each term of the sum one can nd its amplitude a k (t) (instantaneous power) and instantaneous frequency ω k (t) = ϕ k (t). The EMD method provides an algorithm able to expand the wide class of signals into a sum of simpler, mono-component signals. 14 Quite often, intrinsic modes have a clear interpretation and represent physical modes that arise in the process being investigated. That is what makes EMD method particularly attractive, despite the low maturity of its underlaying mathematical foundations. A time dependent power spectrum built up of instantaneous frequencies calculated for a set of intrinsic mode functions is called a Hilbert-Huang spectrum. Let us remind here, just the denition of imf function: Definition 1 An intrinsic mode function imf(t) must fulll two conditions: 1. the number of its zero crossings points and extrema may dier at most by one; 2. imf(t) needs to be symmetric with respect to its local mean value which must tend to zero at any point. 3 September 2-24, 29

4 The EMD method has been already successfully applied to study HET plasma oscillations measured by 6, 1922 an array of antennas and electric probes galvanically coupled to plasma. In those papers one can nd more details about the calculation of analytical signals and instantaneous frequencies, the construction of Hilbert-Huang and marginal power spectra as well as determination of phase locks and the identication of possible error sources. It is known that one of the weak points of EMD method is mixing signicantly dierent time scales in the same mode. It is an inherent feature of the original algorithm 14 which in the case of intermittent signals may generate ctitious variations in the resulting imfs and, hence, in the instantaneous frequency values. The possible way of avoiding this shortcoming consist on so-called ltering of intermittences. 16 It is based on the wavelength criterion which moves too slow oscillations identied in the current mode to the subsequent imf(t) (or even to several subsequent modes the routine is an iterative one). It is very eective procedure providing that a proper smoothing is applied at the boundaries of the cut-o (moved to the next imf(t)) sequences of oscillations. 6 However, introducing somehow arbitrary wavelength to dene the criterion EMD partially loses its self-adaptive character which is perhaps the most valuable feature of the method. To keep the self adaptive nature of EMD the other solution has been suggested (see Ref. 17). It was indicated that EMD operates like a bank of dyadic lters if the signal represents uniformly distributed white noise. 23 It means that all time scales of white noise (not closer in frequencies than an octave) will be ltered out into the separate modes. However, any signal of intermittent character may be enriched in the lacking time scales by an addition of uniform white noise of nite amplitude. Applying EMD to such enriched signal one can signicantly reduce time scale mixing. The addition of white noise can be nally canceled by averaging each particular imf j (t) over the big number of decompositions obtained with dierent white noise realizations. This noise assisted EMD method is called ensemble empirical mode decomposition 17 (EEMD). The EEMD method generates the other issue: the modes averaged over the ensemble may be not an intrinsic mode function in the sense of Denition 1. Usually they are not symmetric with respect to zero and thus both the points of the denition may be violated. However, it may be corrected by application of EMD again, now to the particular averaged mode. Anyway, the improvement introduced by EEMD is so signicant that we have decided to apply this method for processing of the signals captured in the current experiment. III. Experimental arrangement PPSXML HET was operated in the PIVOINE ground-test facility. 24 Two series of signals were captured: for the rst one discharge voltage was varied from 2 to 8 V (keeping Xe ow rate at the level of 6 mg/s and magnetic eld coil current I c =17 A) while for the second one from 2 to 1 V (reducing Xe ow rate to 4 mg/s and increasing coil current I c to 2 A). In several runs coil current was intentionally adjusted to gain more stable discharge regime. Discharge current I D as measured by Tektronix current probe at the anode side of the thruster was always used as a main reference characteristics. A set of single cylindrical Langmuir probes each shielded with cylindrical screen placed between two concentric alumina isolators was used. Probe tips were made of tungsten wire with the diameter 2R p =.1 mm and the length L = 5 mm. Heat load to the probe was estimated assuming average electron temperature of 16 ev and average electron density n e = 1 16 m 3 (keeping in mind that the probes operate in the shadow of the isolator). In such a way the upper limit of save operation was established up to the polarizing voltage of +25 V. To capture probe signals and bias the probes with constant voltage we used a conventional electric circuit with a oating battery and measuring resistor of 1 kω. The one end of this resistor was connected to Tektronix TDS514 oscilloscope with the use of 5 Ω concentric signal cable while the opposite end of it was grounded. The signals were captured when probes were biased with voltage of +19 V or when the batteries were disconnected at all. The probe signals were stored in the memory of the oscilloscope operating in DC mode. To avoid frequency band limitation the concentric cables could be terminated with 5 Ω resistors at the oscilloscope side however, in the frequency range of our interest, it was not necessary. Moreover, being interested in waveforms of MF range we always used 2 MHz oscilloscope lters. On the other hand operating to the open input of the oscilloscope resulted in the signicantly higher signals ensuring still sucient frequency resolution while signal to noise ratio was improved. The sampling frequency was 25 MHz while the length of recorded time series was 5 ksamples (2 µs). The probe axes were always directed along the radius of the thruster and located in the geometric shadow 4 September 2-24, 29

5 of the outer isolator; two dierent probe congurations were applied: two pairs of probes located downstream of the thruster exit in z=3.7 ( of the rst pair and C-probe of the second pair) and 13.7 mm ( of the rst pair and D probe of the second pair) the ends of the probe tips were positioned of about 7 mm from the inner circumference of the outer insulator. Coordinate z=. corresponds to the exit plane of the thruster (edge of the outer insulator). Each pair of the probes were azimuthally shifted with respect to the cathode by the angle of ±7. three probes immersed in the 1 mm wide slit that had been cut in the outer isolator of the thruster in this case the probe z-coordinates were (), 3.8 () and 8.8 mm (C-probe). The ends of probe tips were aligned with the internal wall of the insulator. Figure 1. Electric probes used in the experiment: a) pairs of A-B and C-D probes positioned downstream of the exit plane, ±7 with respect to the cathode (probe tip distance in each pair equals 1 mm), b) A, B and C-probe in a 1 mm wide slit of the HET isolator. Generally speaking, to measure plasma parameters the specic probe theory that corresponds to the conditions of the experiment must be applied. For the discussion of probe theories the reader is referred to one of the monographes, for instance Ref. 25 and the bibliography cited in the newer papers e.g. Ref. 26. Capturing current-voltage (CV) characteristics of the probe it is possible in principle, to determine n e, n i, T e, plasma potential V s and often electron energy distribution function EEDF. Thus, current to the probe is a function of several plasma parameters and voltage that is used for polarization of the probe tip with respect to plasma. In the case of HET plasma there is no doubt that probes of sub-millimeter diameters operate in the collisionless limit in which mean free path: λ e,i >> R p. However, due to variation of plasma parameters in dierent regions of its volume the variable ratio of sheath thickness to probe radius referred to as Debye parameter ξ p = R p /λ D shifts its operation regime from thin to thick sheath limit (OML theory) when ξ p. For instance, for our probes ξ p varies in the range of when n e decreases from 1 18 m 3 to 1 16 m 3 providing constant T e = 16 ev. IV. Results and discussion To make the measurements as simple as possible we have biased all the probes with the same constant voltage (+19 V) or kept them close to the ground potential. In such a way we operated the probes nearby to the knee of ion saturation current. Nevertheless, the probes usually collected both currents with excess of electrons (when the probes were unbiased at all we also observed periods of strictly positive current for HET operated at the lowest voltages see Figure 2 b). In such a way the uctuations of probe currents have to reect the superposition of several plasma parameter uctuations. The time of sheath formation t s was estimated by the expected ion plasma frequency which for n i varies from t s.5µs to.5µs respectively. The oscillations with characteristic time scale lower than t s were discarded from the analysis. 5 September 2-24, 29

6 I D - amps U A - V U B - V U C - V U D - V R m =1 kω; U D =2 V, I C =17 A, dm/dt=6 mg/s I D - amps U A - V U B - V U C - V U D - V R m =1 kω; U D =7 V, I C =17 A, dm/dt=6 mg/s Figure 2. Raw signals as captured at xenon ow rate of 6 mg/s and coil current of 17 A when A-B and C-D probes were unbiased and positioned downstream of the PPSX exit plane a) U D = 2 V, b) U D = 7 V. Nonstationary signals representing uctuations of discharge and electric probe currents were studied using their raw records as well they were expanded into nite sets of intrinsic mode functions imf's with the use EEMD (or EMD) method. The analysis is applied to the signals captured in dierent operating conditions as it was depicted in Section III. A. A-B and C-D probes downstream of the PPSX HET exit plane 1. 6 mg/s xenon ow rate, 17 Amps coil current Comparing waveforms of discharge and probe currents as captured at the low and high discharge voltages (here at U d = 2 and 7 V) there is a qualitative dierence in their shape. The corresponding raw signals are presented in Figure 2. It appears that at U d = 2 V the thruster is operated close to the knee of its I-V characteristics. Here the mean value of discharge current < I d >= 3.42 A and its standard deviation σ = 1.11 A. The breathing mode is very slow ( 8 khz) and well replicated by the waveform of each probe current. The same behavior was observed for each correlated signals that were recorded in the same operating conditions. Applying EEMD method to discharge current and all the signals captured by the probes the corresponding sequences of intrinsic mode functions were generated (usually 15 modes per signal). Most signicant modes extracted from the signals of A and B probes (here imf 9 imf 13 ) are reproduced in Figure 3. Their power spectra with frequencies varying in the range of 3 to 2 khz make some of them to be consider as a representative of transit time oscillations. Calculating cross-correlation functions for the corresponding modes imf 1, imf 11 and imf 12 one can nd that their maximum values xc AB equal to.84,.79 and 6 September 2-24, 29

7 .9 while the time delay values AB equal to.,. and.8 µs respectively. However, the characteristic frequency of the mode imf 12 fused in the vicinity of 3 khz rather exclude it from the family of transit time oscillations. On the other hand time delay AB =.8 µs at the A-B probe distance of 1 mm results in the possible phase velocity v ph m/s. The appropriate modes of probes C and D results in the maximum values of cross-correlation functions smaller of about.1 than in the case of A-B probes with time delay CD.5µ. imf 9 A - a.u. imf 1 A - a.u. imf 11 A - a.u I d - a.u. imf 9 A - a.u. imf 1 A - a.u. imf 11 A - a.u I d - a.u. 8 8 imf 12 A - a.u imf 12 A - a.u imf 13 A - a.u imf 13 A - a.u Figure 3. EEMD of probe signals as captured at U D = 2 V, xenon ow rate of 6 mg/s and coil current of 17 A (unbiased probes positioned downstream of the PPSX exit plane) a), b). In the central panel discharge current I d is also depicted. Only the most signicant modes and appropriate residua are shown. Raw signals are in green. Increasing discharge voltage to 3 V the mean value of discharge current < I d >= 3.3 A and its standard deviation decreases to σ =.23 A. In this operating conditions the uctuations of the probe signals become indeed more signicant than previously. However, for the MF modes maxima of cross-correlation function xc AB.85 are slightly lower than in the case of U d = 2 V while the resulting value of time delays AB.2 µs seems to be below the accuracy of the measurements. In Figure 4 modes generated by EEMD applied to signal as well as to discharge current are shown for comparison. The frequency content of the appropriate modes is similar. However, the correlation of probe and discharge current is weaker than it was observed in the case of two probe signals. For instance, now xc AI.42 for modes imf 9 and increases to xc AI.65 for modes imf 11. What is interesting there is no time delay AI between modes imf 9, while for the modes imf 11 it increases to 3 µs. The further growth of discharge voltage, results in deeper modulation of discharge current with well separated local extrema. And thus for U d = 7 V the mean value < I d >= 3.86 A and standard deviation increases to σ =.85 A. The main modes resulting after application of EEMD are shown in Figure 5. Calculations of the cross correlation functions for the most signicant MF modes, mainly imf 9 family, 7 September 2-24, 29

8 12 1 imf 13 A - a.u. imf 12 A - a.u. imf 11 A - a.u. imf 1 A - a.u. imf 9 A - a.u I d - a.u imf 13 A - a.u. imf 12 A - a.u. imf 11 A - a.u. imf 1 A - a.u. imf 9 A - a.u A - a.u Figure 4. EEMD of signal as captured at U D = 3 V, xenon ow rate of 6 mg/s and coil current of 17 A (unbiased probes positioned downstream of the PPSX exit plane) a), b) discharge current I d. Only the most signicant modes and appropriate residua are shown. Raw signals are in green. result in xc AB =.82 for the A and s (see Figure 6), xc AI =.62 for the and I d, and xc BI =.52 for the and I d. The respective values of time delay AB.5 µs, AI.9 µs and BI 1.6 µs. The frequency of 85 khz is dominating in the Fourier power spectra of the correlated here intrinsic mode functions imf 9. One can wonder if the obtained values of time delay AB and frequency band of imf 9 modes may support the hypothesis that transit time oscillations are longitudinal traveling waves. For the A and s the resulting phase velocity could be v ph m/s what is comparable with the expected speed of ions at the exit of the thruster. However, the values of time delay resulting from the crosscorrelation function can be useful only for harmonic oscillations. Even with visual inspection it is obvious that our modes do not uctuate with well dened frequency. Moreover they are not mono-component signals. Although it is evident that for the family of signals captured in the current experiment the EEMD method works much better than the standard EMD (signicantly reducing mixing of time scales in each mode), the resulting modes only approximate intrinsic mode functions in the sense of the Denition 1. As follows there is no doubts that generated decomposition is good enough to study the correlations. However, the farther decomposition is necessary to study phase relations of the signals. To obtain mono-component modes that strictly satisfy the conditions of Denition 1 classic version of EMD was applied subsequently, after the use of EEMD. Only the modes that could be representative for transit time oscillations were proceeded. As an example the most signicant modes extracted from imf 9 7 V of A and are depicted in Figure 7. Now, they are true intrinsic mode functions and their Hilbert transform can be used for calculation of the phases ϕ(t) and instantaneous frequencies ν = 1 2π dϕ(t) dt. The 8 September 2-24, 29

9 imf 8 A - a.u imf 8 A - a.u imf 9 A - a.u imf 9 A - a.u imf 11 A - a.u. imf 1 A - a.u I d - a.u. imf 11 A - a.u. imf 1 A - a.u A - a.u. 8 8 imf 12 A - a.u imf 12 A - a.u Figure 5. EEMD of signal as captured at U D = 7 V, xenon ow rate of 6 mg/s and coil current of 17 A (unbiased probes positioned downstream of the PPSX exit plane) a), b) discharge current I d. Only the most signicant modes and appropriate residua are shown. Raw signals are in green. xcc AB (t) - a.u time delay - µs PSD(imf 9 ) - a.u frequency -- khz Figure 6. Cross correlation function a) and Fourier power spectrum b) of modes imf 9 extracted from probe and discharge signals as captured at U D = 7 V, xenon ow rate of 6 mg/s and coil current of 17 A (unbiased probes positioned downstream of the PPSX exit plane). wavelength' of each mode varies from cycle to cycle. Neglecting the instantaneous frequency variations per each cycle its slow changes in a longer scale can be correlated with the breathing mode of I d see the smoothing curve in Figure 7. On the other hand the new modes of A and B probes are well synchronized. The appropriate waveforms reproduced in Figure 8 b indicate just small forth and back shifts of separate wiggles from cycle to cycle however, in average the both modes overlap each other. Moreover, the phase dierence ϕ A (t) ϕ B (t) only slightly uctuates close to the constant value (here zero or -2π). The presence of these uctuations and lack 9 September 2-24, 29

10 imf 9 A - a.u. ϕ(t) A - rad (dϕ(t)/dt) A - khz smooth 145 khz imf 9 of EEMD imf 1 of additional EMD imf 9 B - a.u. ϕ(t) B - rad (dϕ(t)/dt) B - khz smooth 145 khz imf 9 of EEMD imf 1 of additional EMD Figure 7. The main mode resulting after subsequent application of EMD to imf 9 already generated by EEMD additionally phase variation per cycle and instantaneous frequencies are shown; a), b) B- probe. U D = 7 V, xenon ow rate 6 mg/s, coil current 17 A (unbiased probes positioned downstream of the PPSX exit plane); ϕ(t)/2π *x *x ϕ A (t) - ϕ B (t) ϕ A - ϕ B Figure 8. a) ϕ A (t)/2π and ϕ B (t)/2π of the main modes resulting after subsequent application of EMD to the respective imf 9 s already generated by EEMD additionally straight line ts are shown; b) the corresponding phase dierence ϕ A (t) ϕ B (t) superimposed on the respective imfs. U D = 7 V, xenon ow rate 6 mg/s, coil current 17 A (unbiased probes positioned downstream of the PPSX exit plane). of non-negligible trend may be the argument testifying the volumetric character of the examined modes mg/s xenon ow rate, 2 Amps coil current The reduction of mass ow rate to 4 mg/s allows to operate PPSX thruster up to 1 V. However, the correction of magnetic eld is necessary to ensure the stable and safe regime of the discharge. In these operating conditions the correlations between I d and the probe signals (or appropriate imfs and their residua) calculated at dierent values of U d indicate similar behavior as in the case of 6 mg/s xenon ow rate. It means that even for most signicant MF imf s time delay resulting from calculation of correlation functions usually does not implicate the phase lock with the no-negligible phase dierence (modulo 2π) that exceeds the noise level. However, the signals corresponding to each value of discharge voltage have to be considered individually. For instance, while signicant MF component (centered about 19 khz) is unambiguously present in I d signal captured at U d = 4 V, its correlation with corresponding modes of probe currents is very poor for imf 9 maximum xc AI =.1 at AI =.25µs while for imf 1 maximum xc AI =.4 at extremely long AI = 12µs see Figure 9 a and 11 a. Thus, the MF components contained in the discharge current are de-correlated 1 September 2-24, 29

11 I D - amps U A - V U B - V U d =4 V I D - amps U A - V U B - V U d =1 V U p =+19 V U p =+19V Figure 9. Discharge and probe current signals as captured for xenon ow rate of 4 mg/s and coil current of 2 A: a) U d = 4 V, unbiased probes b) U d = 1 V, probes polarized with +19 V; (probes positioned downstream of the PPSX exit plane). PSD(imf 9 ) - a.u PSD(imf 9 ) - a.u PSD(imf 1 ) - a.u frequency -- khz PSD(imf 1 ) - a.u frequency -- khz Figure 1. Fourier power spectra of the main modes resulting after application of EEMD to discharge and probe currents as captured for xenon ow rate of 4 mg/s and coil current of 2 A: a) U d = 4 V, unbiased probes b) U d = 1 V probes polarized with +19 V; (probes positioned downstream of the PPSX exit plane). with the similar components identied in the probe signals. It is even more evident at the highest discharge voltages see Figure 9 b and 11 b here maximum xc AI =.54 at AI = 1.2µs for imf 9 while maximum xc AI =.15 at AI = 1.4µs for imf 1 (tentative frequency content of particular modes is illustrated with Fourier power spectra reproduced in Figure 1). Moreover, in the considered operating conditions the crosscorrelation functions calculated for MF imfs of the A and signals do not result in signicant time delay: e.g. for U d = 4 V maximum xc AB imf9 =.68 at AB =.6µs and maximum xc AB imf1 =.79 at AB =.8µs. However, even though the calculated time delay AB exceeds the measurement accuracy (estimated here to ±.5µs) the phase and instantaneous frequency analysis usually dose not indicate phase lock intervals longer than just few cycles (two or three) for instance it is the case of U d = 1 V for which maximum xc AB imf9 =.59 at AB =.3µs and xc AB imf1 =.73 at AB =.7µs. The synchronization of appropriate MF modes obtained with EEMD for dierent probe locations in general is better, when the thruster is operated at high voltage. 11 September 2-24, 29

12 imf 9 - a.u imf 9 imf 1 imf 9 - a.u imf 9 imf imf 9 A - a.u imf 9 A - a.u imf 9 B - a.u imf 9 B - a.u imf 1 A - a.u imf 1 A - a.u imf 1 B - a.u imf 1 B - a.u Figure 11. The main modes resulting after application of EEMD to discharge and probe currents as captured for xenon ow rate of 4 mg/s and coil current of 2 A: a) U d = 4 V, unbiased probes b) U d = 1 V, probes polarized with +19 V; (probes positioned downstream of the PPSX exit plane). The residua of each mode are superimposed on the raw signals (in green). B. Three probes in a slit of the PPSX HET isolator The analysis of signals that were captured by the probes when immersed in a 1 mm wide slit of the thruster isolator was done with the use of EEMD method like in the previous case. Similarly, the additional sifting with classic EMD method was applied when detailed phase analysis was necessary. The probe conguration was described in Section III here it is enough to remind that was located in the exit plane of the thruster while B and C probes were positioned along z-axis, upstream of this plane see Figure 1 b. As before, this analysis was eventually expected to point to the the propagation mode and to the nature of MF band oscillations. Even by visual inspection (see Figures 12 a), one can notice that signals recorded with the use of located in the exit plane replicate almost all uctuations of the main discharge current for low values of U d (like for the case of two probes located downstream of the exit plane). The signals captured by B and C-probe usually contain faster MF components that may be hard to detect in I d signal. In the case of high discharge voltage the correlation of signal and I d becomes poor in the examined MF range (see Figures 13 a). It is conrmed by EMD phase analysis and calculations of correlation functions corresponding to the particular modes. For instance, considering family of imf 9 modes (that represent the most intensive component of MF band) the maximum value of cross-correlation function xc BC =.85 at BC =.3 while xc AB =.48 µs at AB =.8 µs and maximum xc A =.17 at A = 8.6 µs. 12 September 2-24, 29

13 I C - ma I B - ma I A - ma I dis - A V, 6 mg/s z = mm; U p = z = -3.8 mm; U p = z = -9 mm; U p = ν(t) - khz ϕ(t) - rad ϕ(t)/2π imf 9 - a.u C-probe C-probe A-C B-C A-B A-I d C-probe Figure 12. Three probes in the slit of the thruster insulator phase analysis of the most intensive MF mode imf 9 for U d = 3 V a) raw signals, b) starting from top: superimposed modes excluded from B and C-probe signals; phase ϕ(t) for imf 9 of A, B and C-probes as well as for I d signals; ϕ(t) calculated for the A-B, B-C, A-C and A-I d combination; corresponding instantaneous frequencies ν(t) (smoothed). Xenon ow rate 6 mg/s and coil current of 17 A; unbiased probes. I C - ma I B - ma I A - ma I dis - A 6. 6 V, 6 mg/s z = mm; U p = z = -3.8 mm; U p = z = -9 mm; U p = ν(t) - khz ϕ(t) - rad ϕ(t)/2π imf 9 - a.u C-probe C-probe A-C B-C A-B A-I d C-probe Figure 13. Three probes in the slit of the thruster insulator phase analysis of the most intensive MF mode imf 9 for U d = 6 V a) raw signals, b) starting from top: superimposed modes excluded from B and C-probe signals; phase ϕ(t) for imf 9 of A, B and C-probes as well as for I d signal; ϕ(t) c for the A-B, B-C, A-C and A-I d combination; corresponding instantaneous frequencies ν(t) (smoothed). Xenon ow rate 6 mg/s and coil current 17 A; unbiased probes. The cross-correlation of appropriate residua of the intrinsic mode functions extracted from probe signals with the EMD method unambiguously shows the existence of a time shift, which can also be recovered by cross-correlation of the raw signals. However, reliable time delays were detected only when the HET was operated at the highest discharged voltages. The phase analysis of MF modes indicates a number of time intervals for which phase lock is evident 13 September 2-24, 29

14 when signals of dierent probes are examined. The results of this analysis are illustrated in Figure 12 b for U d = 3 V and 13 b for U d = 6 V. The most intensive MF modes (here again referred to as imf 9 family) are considered. Despite some imperfectness of the EEMD-EMD procedure the synchronizations of signals captured by B and C-probes is maintained in the intervals lasting often longer than several cycles. However, the resulting phase lock does not imply the measurable phase shift please, note the plateaus in Figures 12 b and 12 b. Thus, the obtained phase shift values that are close to zero (modulo 2π) cannot conrm the propagation of traveling waves. Moreover, the periods of phase lock for A and become shorter than in the case of B and C-probe and corresponding phase dierence starts to uctuate indicating some loss of synchronization between inner and exit zone of the discharge (the synchronization between corresponding imf 9 modes of I d and signal is even worse). It is worth to note that considered MF mode is perhaps induced at the beginning of the acceleration zone of the discharge. The phase lock of discussed here imf 9 modes seems to be maintained not longer then the cycle of the breathing mode the slower breathing mode uctuations the longer phase lock intervals. It is worth to note that for the most operating conditions we could measure the probe current uctuation above the noise level only if 1 kω measuring resistor was not terminated with 5 Ω (our line impedance) at the oscilloscope side. Thus, the limitation of reliable correlation analysis is determined by RC constant of the cabling system here.4µs. On the other hand, analyzing probe signals that collect simultaneously electron and ion current we have to keep in mind that the inuence of plasma potential uctuations may be indeed signicant at the increasing part of the probe CV characteristics. 25 However, as was suggested in Ref. 13, the oscillations of plasma potential display essentially a volumetric character. Consequently the eect of longitudinal propagation that is linked to ion density uctuations may be masked in the probe signals by volumetric uctuations of plasma potential. To reduce the inuence of plasma potential uctuation it is necessary to ensure the probe operation in the ion branch of its CV characteristics. The better time resolution is possible by impedance matching with the use of proper preamplier 4 V. Conclusion Plasma oscillations of 1-5 khz band are studied experimentally in PPSXML high voltage thruster. The HET operating conditions are varied in the wide range. The correlations of signals obtained for probes located along the z-axis of the thruster are analyzed. Langmuir probe tips are directed along the thruster radius and located in the shadow of the outer insulator. Two probe congurations are investigated: a) two pairs of probes (A-B and C-D) located downstream of the thruster exit and b) three probes immersed in a 1 mm wide slit of the outer isolator of the thruster The level of the studied MF range oscillations (referred to as transit time oscillations) is found to be greatly enhanced when the HET operates in the HV regime. The cross-correlation analysis was eventually expected to point to the the propagation mode and to the nature of MF band oscillations. Consequently, the main goal of the measurements was to check whether a phase shift (or time delay) exists between the signals collected at dierent z-locations (provided that the relative signals are synchronized), or if the oscillations measured locally have essentially a volumetric character and are thus related to a standing wave. Both the EEMD-EMD methods along with Hilbert-Huang transform and classic correlation analysis have been used to assess the signal coherence. In conclusion, the phase analysis of MF modes (extracted from the raw signal with the use of EEMD-EMD method) indicates a number of time intervals for which phase lock is evident when signals of dierent probes are examined. However, the observed synchronization of MF mode (with the same instantaneous frequency) does not imply the measurable phase shift. Moreover, the evolution of the signals waveform downstream of the thruster does not allow us to state that the collected MF oscillations are indeed associated to a traveling wave. A denite answer to this question would require a better spatial and time resolution (more probes in the slit of the isolator) in order to accommodate for the axial displacement of the excited region under dierent operating conditions. Moreover, the probe polarization have to ensure ion current collection in the all probe locations. The new experiment is being prepared. 14 September 2-24, 29

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16 26 McMahon, J. C., Xu, G. Z., and Laframboise, J. G., The eect of ion drift on the sheath, presheath, and ion-current collection for cylinders in a collisionless plasma, Physics of Plasmas, Vol. 12, No. 6, 25, p September 2-24, 29