Discharge and ion current
A typical VAT discharge current trace is shown in Fig. 3. The t=0 μs time corresponds to the opening of the PJP capacitor bank for energy release into the circuit. The current pulse lasts about 25 μs. The current rises up to 4 kA in less than 10 μs although the input power is only 30 W. Such a large current indicates the thruster generates a very large instantaneous thrust. Assuming the discharge current is entirely converted into singly-charged ion current, the thrust is above 10 N at the peak. Note that as the current transformer is placed upstream of the cathode, see Fig. 2, the fraction of current that goes through the trigger electrode instead of flowing through the anode remains unknown.
Figure 4 shows typical ion current traces observed in the plasma jet with the planar probe. The probe was polarized at -75 V to solely capture ions. As can be seen in Fig. 4, current traces are not strictly identical for different pulses although the general shape stays unchanged. This illustrates the fact that vacuum arcs are complex physical phenomena, yet with a certain level of reproducibility and stability. The arc starts to form where the asperities of the cathode give locally a larger electric field, initiating a cathode spot through an explosive phenomenon [4]. As cathode spots have a limited lifetime [6], the arc moves or jumps to other locations on the cathode surface where the electric field is high, initiating other cathode spots. Thus, for a similar discharge current distribution through the cathode, the resulting plasma discharge and plasma jet are different from one pulse to another. However, in the case of the PJP thruster, it is quite common to observe two well-distinct peaks on the ion current profile in the plasma jet for a Ni/Cr cathode, as exemplified in Fig. 4.
Langmuir probe
Acquisition of a Langmuir probe I-V characteristic curve is usually performed by smoothly sweeping the probe bias voltage while recording the current [7,8,9,10,11]. The analysis of the measured current against applied voltage curve gives access to the local electron properties and to the local plasma potential [7]. In order to obtain the temporal evolution of a Langmuir probe I-V curve, the voltage frequency sweep has to be much larger than the typical frequency of the observed phenomenon or process. According to waveforms displayed in Figs. 3 and 4, the frequency must be above 1 MHz in this study, which makes measurements extremely complicated for a single shot from a technical viewpoint [12, 13]. Moreover, the evolution of a vacuum arc is fast and stochastic [14], which makes data treatment and analysis challenging. An alternative method is then proposed here to properly recover the change in time of the probe I-V curve in the course of the VAT discharge [15]. This method permits to recover the typical evolution in time of the plasma created during a PJP discharge based on an indirect, or average, approach instead of a direct approach.
The Langmuir probe voltage has been gradually increased from -30 V to +50 V. In order to increase the amount of data and the accuracy, the voltage step was set to 0.5 V between -5 V and 10 V and between 20 V and 30 V. The stepsize was set to 1 V elsewhere. For each voltage step, 20 current traces were recorded and then averaged as the waveform noticeably vary from one pulse to another. The capacitor bank trigger signal is used as a time reference for all recordings. Figure 5 shows the evolution of the mean current trace for various bias voltages. A negative current corresponds here to ions, while a positive current indicates that electrons are captured. As expected, ions are collected when the probe voltage is negative with respect to the ground (reference) while electrons are collected for positive voltages. Besides, the electron current is larger than the ion current in agreement with the Langmuir probe theory. Note that at t=0, i.e. at thruster ignition, a very-high frequency electromagnetic perturbation is always captured by the probes, see Fig. 5.
Once all waveforms have been acquired over the selected voltage range, a home-made Python program converts the time–current–voltage matrix into a current–voltage–time matrix. In other words, the evolution in time of the probe current for a given voltage is replaced by the change in current as a function of the applied voltage for a given time, or in short the I-V curve for a given time [15]. Figure 6 shows the reconstructed I-V curve at 16.5 μs and 28 μs. As expected the Langmuir probe characteristic depends on the point in time when measurements are performed in the course of the current pulse.
The Langmuir probe analysis is based on the recovery of physical quantities through analytical properties of the Intensity vs Voltage (I-V) curve. Models have been developed to fit experimental data, in order to link properties of the I-V curve to physical properties. The most widely used theory for LP analysis is the Orbital Motion Limited theory (OML). Originally developped by Langmuir and Mott-Smith in 1926 [16], this model assumes an infinite radius of the sheath surrounding the probe. It comes that the ions are attracted from a large distance. A large sheath often implies a low plasma density, and the OML theory should be inaccurate for collisional plasmas. However, Chen showed experimentally in [17] its “surprising” validity for high density plasmas. This theory is popular thanks to its computational simplicity leading to consistent results. Data analysis is indeed based on successive linear fits on the I-V curve, as the \(I_{e}^{2}\) vs V slope is proportional to \(n_{e}^{2}\).
We however wanted in this study to implement models that takes into account more parameters, in particular a better sheath theory. For achieving this goal, Poisson’s equation should be solved for the potential V(r) that goes from the probe surface to infinity. Chronologically, a theory was first developed in 1957 by Allen, Boyd and Reynolds (ABR) for spherical probes [18] (later extended to cylindrical probes by Chen [19]). ABR theory, although solving the Poisson’s equation, assumes that Ti=0, which would mean that the ions are attracted radially to the probe without any initial velocity.
In reality, though, ions present a finite Ti and are only collected by the probe when their angular momentum is small enough, due to orbital motion around the probe. In order to solve this problem, Bernstein and Rabinowitz developped in 1959 a theory [20] that takes into account a finite ion temperature, assuming mono-energetic ions. Laframboise later extended their work by making the model parametrisation for Maxwellian distributions [21]. This so-called BRL (Bernstein-Rabinowitz-Laframboise) model takes then into account the ion angular momentum, as well as the existence of an absorption radius around the probe. Depending on the probe potential and on the ion angular momentum, some ions are trapped and collected by the probe when they cross this virtual boundary. Hence, the BRL is the most complete theory presented here.
Concerning the electron temperature, the ABR and BRL models give very similar values as they both assume a Maxwellian energy distribution function for the electrons. The electronic part of the I-V curve is then an exponential function between the floating potential and the plasma potential Vp. The electron temperature is determined from the slope of the natural logarithm of the electron current in the transition region: \(T_{e} = \left (\frac {dln(I_{e})}{dV} \right)^{-1}\).
The plasma potential Vp can be inferred from the ABR and BRL models. However, the two models provide a value for the ion branch and another one for the electron branch. The two values of Vp do not match under our conditions due to the fact that the electron density is relatively large [7, 22]. For that reason, Vp is classically defined here as the voltage for which the first derivative of the I-V curve is the highest. Vp indicates the inflection point of the curve beyond which the electron current saturates [8,9,10].
Examples of calculation outcomes are presented in Table 2 based on the I-V curves obtained at 16.5 μs and 28 μs, see Fig. 6. The BRL theory always gives an electron density larger than the ABR theory, as observed in the past with low-pressure radio-frequency discharges [22]. The OML model is most of the time in good agreement with the BRL model in this work. The electron temperatures are almost identical as the same approach is used in the two models. The slight difference could be due to the substraction of the ion current.
Table 2 Plasma parameters comparison at two different instants