Design and fabrication of a full elastic submicron-Newton scale thrust measurement system for plasma micro thrusters

Zhongkai ZHANG(张仲恺),Guanrong HANG(杭观荣),Jiayun QI(齐佳运),Zun ZHANG (张尊), Zhe ZHANG (章喆), Jiubin LIU (刘久镔),Wenjiang YANG (杨文将) and Haibin TANG (汤海滨),5,6

1 School of Space and Environment, Beihang University, Beijing 100191, People’s Republic of China

2 Shanghai Institute of Space Propulsion, Shanghai Engineering Research Center of Space Engine,Shanghai 201112, People’s Republic of China

3 School of Astronautics, Beihang University, Beijing 100191, People’s Republic of China

4 The Institute of Space Systems, University of Stuttgart, Stuttgart D-70569, Germany

5 Key Laboratory of Spacecraft Design Optimization & Dynamic Simulation Technologies, Ministry of Education, Beijing 100083, People’s Republic of China

6 Laboratory of Space Environment Monitoring and Information Processing, Ministry of Industry and Information Technology, Beijing 100083, People’s Republic of China

Abstract In this work, a force measurement system is proposed to measure the thrust of plasma microthruster with thrust magnitude ranging from sub-micro-Newtons to hundreds micro-Newtons.The thrust measurement system uses an elastic torsional pendulum structure with a capacitance sensor to measure the displacement,which can reflect the position change caused by the applied force perpendicular to the pendulum axis.In the open-loop mode, the steady-state thrust or the impulse of the plasma micro-thruster can be obtained from the swing of the pendulum,and in the closed-loop mode the steady-state thrust can be obtained from the feedback force that keeps the pendulum at a specific position.The thrust respond of the system was calibrated using an electrostatic weak force generation device.Experimental results show that the system can measure a thrust range from 0 to 200 μN in both open-loop mode and closed-loop mode with a thrust resolution of 0.1 μN, and the system can response to a pulse bit at the magnitude of 0.1μN s generated by a micro cathode arc thruster.The background noise of the closed-loop mode is lower than that of the open-loop mode, both less than 0.1/μN Hz in the range of 10 mHz to 5 Hz.

Keywords: plasma thruster, micro-thrust measurement, torsional pendulum

1.Introduction

In recent years, more and more plasma propulsion devices(electric thrusters) have been widely used in various applications such as orbit keeping, drag force compensation, altitude control and adjustment,and active deorbiting[1–3].Due to the high specific impulse and the ability to precisely control micro thrust, electric thrusters, especially micro electric thrusters, have better adaptability than traditional chemical thrusters and cold gas micro thrusters, and can provide more options for diverse space missions[2].Based on the operating principle, electric thrusters can be classified as electrostatic,electromagnetic and electrothermal.Various specific micro plasma thrusters, such as low-power Hall thruster, radio-frequency or microwave ion thrusters, field emission electric propulsion (FEEP), micro cathode arc thruster (μCAT),pulsed plasma thruster (PPT) and other technologies either working in steady state or pulse mode [4–11] are under continuous research.In general, the propellant of a thruster can be gaseous,liquid or solid.When the propellant is heated,ionized and accelerated, it can gain energy to be discharged from the thruster while momentum exchange takes place.Since micro plasma thrusters produce thrusts ranging from sub-micro-Newtons to sub-milli-Newtons, ground-based thrust measurement systems with high thrust resolution and wide range are urgently needed for more accurate measurement of parameters and characteristics of plasma micro thrusters.

There have been several studies on thrust measurement systems with different ranges and resolutions for thrust ranges in the magnitude of sub-micro-Newtons to mili-Newtons[12].From the classification of measurement principle and device structure,the measurement methods of weak forces of micro-Newton magnitude can be divided into torsional pendulum systems, torsional suspend wires system, single and double pendulum and cantilever type systems, etc to meet different needs of thrust measurement range and resolution.Andrewset al[13] developed a torsional pendulum thrust stand with a gas dynamic calibration device that was capable of resolving thrusts of tens of nano-Newtons to micro-Newtons.Manuelet al[14] used a torsional pendulum measurement device which detected the deflection of the torsion pendulum,and the device was able to measure the impulse thrust of plasma micro thrusters with typical values of a few tens of micro-Newtons, and its control of the system background noise provided the technical support for the Laser Interferometer Space Antenna (LISA) [15, 16] project.A similar torsional pendulum device to measure laser-ablative micro thrusters with the impulse bit of 1–10μN s has been developed by researcher from the University of Tokyo[17].Kurtet al[18]developed a vertical pendulum thrust measurement system and it was capable of measuring micro thrusts of 100μN or more, and its impressive pendulum structure was capable of carrying thrusters with the mass up to 125 kg.Beihang University [19, 20] developed a pulsed thrust measurement system with novel electromagnetic calibration techniques on the magnitude of sub-milli-Newtons earliest in China.Boccalettoet al[21]designed a thrust measurement device with a double-suspending wire structure and measured the thrust of a FEEP thruster, and Yanget al[22] measured the thrust characteristics of a PPT thruster using a similar suspending wire pendulum device in both open and closed operation.Zhouet al[23]developed a torsional pendulum structure with the resolution as low as 24.4 nN,and Soniet al[24]also used a similar torsional pendulum with low spring rate to achieve the resolution of 10 nN.Subhaet al[25] designed a novel compact thrust measurement device to verify a 50μN cold gas thruster using a suspended plate.The Brazilian researchers[26]used a unique electrostatic calibration device for static and dynamic calibration of the torsional pendulum system with tens of micro-Newton in recent years.

At present,there are relatively few measurement systems applicable to plasma micro thrusters with thrust ranges from sub-micron-Newtons to hundreds micro-Newtons, and most of the existing measurement systems cannot achieve the submicron-Newton resolution and the minimum thrust response for thrusters working in either steady state or pulsed state with low background noise applicable to plasma micro thrusters.According to the requirement of the thrust range and resolution, an elastic micro-Newton thrust balance based on torsional pendulum structure is designed.A capacitance sensor is employed to measure the displacement, which can reflect the position change after the force is imposed.Compared with other types of micro-Newtonian thrust measuring devices,this system can operate in open-loop measurement mode or closed-loop measurement mode,where the measured thrust or pulse bit is related to the displacement signal or the feedback signal respectively.When operating in closed-loop mode,there is an additional advantage of eliminating the errors caused by the thruster’s cable, while greatly reducing the noise in the thrust measurement system.In addition, the application of an electrostatic weak force generator in the thrust measurement system calibrates the thrust response of the system by generating a previously known weak force.Furthermore, the pulsed response of a micro cathode arc thruster was evaluated by the thrust measurement system,which validated the practicality of the system in open-loop mode.In summary, the thrust measurement system was validated for the first time in an elastic torsional pendulum using both open-loop and closed-loop methods for steadystate and impulsive forces, achieving measurement ranges in the sub-micro-Newtons to hundreds of micro-Newtons with the ability of 0.1 micro-Newton magnitude and low background noise.

In this work, the dual-modes full elastic thrust measurement system proposed shows credibility and practicability under sub-micro-Newton to hundreds of micro-Newton magnitude in section 2, the structure of the thrust measurement system and the weak force generation device will be presented in detail, while in section 3, the steady thrust response and characteristic of the system will be discussed with the dynamic response of a micro cathode arc thruster.Finally, the conclusion will be given in the final part.

2.Construction and principle of the thrust measurement system

The image and the schematic diagram of the thrust measurement system are shown in figure 1.The system is composed of the rigid frame,the torsional pendulum of the thrust stand, a displacement sensor and a weak force generation device.The torsional pendulum is connected by two vertically placed shafts and is connected with the rigid frame consequently.The plasma micro thruster is installed at one end of the torsional pendulum, and the counterweight is installed at the other end accordingly.The position limit frame is used to limit the rotation and pitch angle of the axis.The capacitive displacement sensor with the resolution of 40 nm is used to measure the position change of the torsional pendulum, and the micro-Newton weak force is generated by an electrostatic device attached to the torsional pendulum.A magnet sheet is fixed on the other side of the pendulum, and the current generated by the feedback coil interacts with it to pull or push the torsional pendulum in the closed-loop operation.The pipeline and cables of the plasma micro thruster is placed along the pendulum based on the actual demand.

2.1.Open-loop methodology

When the thrust measurement system is operating in openloop mode,the pendulum is a regular torsional thrust balance and the principle can be schematically described in figure 2.In figure 2,rexandrsensorare the distance from the axis to the external force point and the displacement sensor respectively, and it is assumed that the external forceFexis perpendicular to the pendulum for convenience.Δxis the displacement signal detected by the displacement sensor,corresponding to the rotation angleθof the pendulum.When the torsional pendulum deviates from the balance position and within a small angle, the restoring moment spontaneously generated by the axis is opposite to the direction of the moment caused by the external force.The theoretical description of the mechanical behavior of the torsional pendulum is well described in the literature [27, 28].Applying Newton’s second law, the result is expressed as [27]:

whereJis the moment of inertia, λ is the total torsional damping factor,kis the torsional spring constant andFexis the external force mentioned above, which can be generated by either micro thrusters or the weak force generation device.

When the torsional pendulum deflects within a small angle, the coefficientsJ, λ andkcan be considered as constants [27].As the deflection angle θ is relatively small, the relationship between θ and displacementΔxcan be written as:

Therefore, the mechanical behavior equation of the torsional pendulum can be written as:

When it is required to measure the steady thrust, the external torque and the restoring moment should be balanced after the torsional pendulum deflecting to a small angle, and the pendulum will finally stay static or slightly oscillate around the equilibrium position whereclose to zero, consequently the relationship betweenFexandΔxcan be simplified as [27]:

The impulse of micro thruster can be measured according to the impulse exchange between the thruster and the torsional pendulum.When the impulse generated by thruster is transmitted to the torsional pendulum, the pendulum will oscillate and the amplitude of the oscillation is related to the torsional spring constant and the moment of inertia of the pendulum.The maximum amplitude of oscillationAmaxis defined as[28]:

HereIbitis the impulse provide by the external force,Jis the moment of inertia andωdis the frequency of damped oscillation defined as follows [28]:

Here the parameterζ[27] is the damping ratio andandωnis the natural frequency of oscillation of the open-loop system.

For an oscillation system such as the torsional pendulum,the oscillation amplitude of the pendulum could be described as [28]:

From the equation (7), the moment of inertia of the torsional pendulum can be acquired by the attenuation envelope of the oscillation curve.

2.2.Closed-loop methodology

When the thrust measurement system is operating in the closed-loop mode, an additional torque is applied to the pendulum to keep it in the same position while being affected by the external torque [29, 30].The detailed principle is schematically described in figure 3.

The feedback closed-loop is composed of the control circuit, feedback electromagnetic coil and permanent magnet sheets.When the feedback loop is activated, the electrical signal received by the displacement sensor is continuously input into the control circuit, and compared with the preset reference voltage in the control circuit.The reference voltage here is the sensor output voltage corresponding to a preset distance between the pendulum and the sensor.When there is a voltage difference between the sensor signal and the reference signal,it means that the current position of the torsional pendulum deviates from the preset position, and the control circuit will output a specific current according to the proportional-integral-derivative (PID) control method [29].The current through the feedback coil will interact with the magnet sheet fixed on the pendulum and generate a certain force to push or pull the pendulum back to the preset position.

The schematic diagram of the PID control circuit used here is shown in figure 1.The output current is composed of the proportional, integral and derivative operation of the difference voltage between the sensor signal voltage and the preset voltage.When there is an external torque caused by the forceFexacting on the torsional pendulum, the converse feedback torque generated by the PID circuit has to be equal to the external torque, which can counteract the effect of the external torque and make the torsional pendulum return to the preset position.In the experiment, by measuring the force of the feedback circuit, or the feedback voltage signal in the feedback circuit, the external force torque can be indirectly measured.

The main advantage of using feedback method is that it can effectively avoid the effect of external influence such as mechanical friction,pipeline and cable restraint in the thruster measurement and eliminate the errors caused by external factors as far as possible.In addition,the closed-loop method can effectively suppress the natural oscillation of the torsional pendulum and eliminate the influence effectively, furthermore, it can reduce the thrust measurement noise and effectively improve the accuracy of steady-thrust measurement.

2.3.The electrostatic weak force generation device

The electrostatic weak force is generated by two electrostatic comb pairs, and the structure is shown in figure 4.

The two electrostatic combs are made of aluminum and have 20 fins and 19 fins respectively.The length and thickness of the fins in this project are 10 and 1 mm and the space between two fins is 2 mm.Both combs are insulated from their holders by PTFE insulator, and an electrostatic comb is fixed on the torsional pendulum while another is installed on a two-dimensional displacement table, which can adjust the relative position of the combs precisely.When there is a voltage between the two combs and fins, the electrostatic attraction force will be generated.According to the literature,the approximate equation of electrostatic force between the two combs is described as follows [31, 32]

HereNis the total number of fins on the combs,candgare the thickness of each fin and the separation between fins respectively, andx0is the engagement distance between fins[32].For the weak force generation device in this experiment,they are constants when the relative position of the combs is determined, and the electrostatic forceFcan be described as:

Firstly, the weak force generated by the electrostatic comb is measured by a high-precision electronic balance with the resolution of 10μg(Sartorius CPA26P series)in advance.The electrostatic combs are powered by an IT6723G auto range DC stabilized power and it is able to provide voltages from 0 to 600 V with the rise time less than half a second.According to the balance reading and the acceleration of gravityg, the relationship between the force of the electrostatic combs and the voltage applied is obtained.For convenience, the force of the electrostatic comb has been transformed into an equivalent external force at the position of the thrusters under the same torque, and the images of the measurement process and the results of the electrostatic comb are shown in figure 5.

In the experiment, the voltage range on the electrostatic combs is 0–550 V with the interval of 50 V, and the horizontal axis is the square of the voltage.The fitting line of the electrostatic force and the square of the voltage is obtained by the least-squares method [33] and the equation is:

Here,a=1.1 ×10−3(μNV−2)andb= − 6 .1 ×10−3(μN) .The linear goodness of fitR2of the slope value is greater than 0.9999 with the error of slopeδ=1.4 ×10−6,and it is suggested that the electrostatic force has good linearity for calibration.The enlarged part of the blue box in the figure 5 is the straight line fitting curve of the corresponding electrostatic force when the voltage ranges from 0 to 100 V.In this voltage range the linear goodness of fitR2is 0.9997 and the electrostatic force is close to the minimum resolution of the electronic balance which is easy to be disturbed by the environment.According to the results of several measurements of force of the electrostatic comb,the error range of the electrostatic force affected by the comb and external environment at different voltages was noted in figure 5.The range of electrostatic forces generated at higher voltages was larger, when less interference was applied from the ambient airflow and the error ratio was near stable according to the data in figure 5.Analysis of the error ratio showed that the calibration process produces an average error of 4.4%.

3.Results and discussion

3.1.Open-loop measurement

The static calibration process can determine the torsional spring rate of the pendulum system.In the open-loop mode of the experiment, the known constant electrostatic force was applied on the torsional pendulum, and the actual deflection angle of the pendulum was determined according to the output signal of the displacement sensor.Therefore, the rotation angle of the axis or the signal voltage change of the displacement sensor should be proportional to the external force.In the experiment, the voltage applied on the electrostatic comb and the corresponding force were from the calibration chart shown in figure 5.When increasing the voltage on the comb pair step by step,the corresponding displacement could be obtained and recorded by a data recorder with a frequency of 1 kHz.

Figure 6 exhibits the displacement data from the sensor when the electrostatic comb pair working voltage increased from 0 to 450 V with the increment of 50 V and then returned to 0 V.The thrusts corresponding to each static thrust step were noted in figure 6,and the thrust values were taken from the calibration chart.

Figure 1.The image and schematic diagram of the thrust measurement system and other experimental setups.

Figure 2.(a) Torsional pendulum at the rest position and (b)torsional pendulum deflected immediately after an impulse was applied to the arm.

Figure 3.(a)Torsional pendulum deflected by the external force and(b) back to the equivalent position by the electromagnetic force of feedback coil.

Figure 4.The image and schematic diagram of the structure of the electrostatic comb.

Figure 5.The images of the measurement process and the linear fit results of the electrostatic force versus the square of applied voltage with error bars.

Figure 6.The displacement data versus time for large spans of applied voltage on the electrostatic comb.

Figure 7.The displacement data versus time for small spans of applied voltage on the electrostatic comb.

Figure 8.The linear fit of the displacement versus square of voltage.

Figure 9.The oscillatory decay curve of the torsional pendulum system and the fitted line of the envelope exponential model.

Figure 10.FFT of natural oscillation of the torsional pendulum.

As shown in figure 6,the high voltage on the electrostatic comb was maintained for 30 s at each step of external force.When the external forces changed, there were obvious and stable steps of the displacement signal after a few seconds.According to the raw data, when the voltage increased or decreased the torsional pendulum oscillated sinusoidally near the position due to the change in applied force, and then the amplitude gradually decreased because of the damping effect.Because the damping factor of the system was relatively small, the torsional pendulum would oscillate near the equilibrium position in a short period of time instead of being completely stationary.If the torsional pendulum was continuously oscillating, the final equilibrium position was the average value of oscillation according to equation (7).

In addition, the output signal of the displacement sensor increased gradually when the external force increased.At the moment when the external force changed, large amplitude oscillation sometimes occurred,such as when the electrostatic comb voltage rose from 150 V to 200 V or 250 V to 300 V.On the contrary, there were small oscillations such as the slight oscillation at the point when the voltage of the comb increased from 300 to 350 V.This was probably due to that the torsional pendulum was not completely stationary at the moment when the external force was applied to the torsional pendulum.When the external force which had a same direction to the angular velocity of the pendulum was applied on the torsional pendulum,there would be an enhanced effect which made the torsional pendulum obtain a greater angular velocity, on the contrary, the direction of the external force was opposite to the angular velocity of the torsional pendulum,there would be an opposite effect and it would reduce the angular velocity of the pendulum after deflecting a certain angle, and the amplitude of the oscillation consequently was relatively small.Furthermore, when the voltage on electrostatic comb decreased step by step after 250 s in figure 6, a certain amplitude of oscillation occurred at the moment exactly when the external force decreased.Besides the above factors, this was also probably due to the release of elastic potential energy stored by the spring.When the torsional pendulum returned to the next equilibrium position with the decrease of the external force, the excess elastic potential energy was released and converted into the kinetic energy,and finally the oscillation of the pendulum was produced.

When applied smaller electrostatic force where the comb voltage ranged from 0 to 100 V and increased with 10 V steps,the relationship between the displacement measured by the sensor and time is shown in figure 7.

As shown in figure 7,the gray line in the figure is the raw data and the red line is the filtered data by Savitzky–Golay filter [34].When the electrostatic comb voltage increased from 0 to 100 V, the measurement point at the pendulum moved about 18 μm approximately due to the electrostatic force.It was suggested by the filtered data that the torsional pendulum could maintain stable at the equilibrium position with the electrostatic force increased and decreased.

In order to evaluate the linearity of the deflection displacement, a linear fit of the displacement versus square of voltage from 0 to 450 V was performed and the results are shown in figure 8.

As shown in figure 8, the red line was the fitting line of the torsional pendulum displacement and square of applied voltage when the voltage of the electrostatic comb ranged from 0 to 400 V.The fitting equation is:

The linear goodness of fitR2of the slope value was greater than 0.9999 with the standard error of slopeδ=4.6 ×10−5,and it was suggested that the displacement had good linearity for calibration.According to equations (10) and (11), the relationship between the displacement and the applied force was obtained by eliminating the square term of voltage:

Equation (12) can be considered as a proportional relationshipF=(k1/k2)Δxand the deviation of the final fitted force wasδF/F=δk1/k1Δx+k1/k2δΔx−δk2/k2Δx.From the relationship between the fitted force and the displacement of the pendulum, and the error was found to be approximately 1.2%.Considering the effect caused by the procedure of the calibration of the electrostatic comb,and the error of open-loop measurement was approximately 5.5%.In addition, according to the straight line fitting model of relationship between deflection displacement and the external force, and the effective force arm of 0.5 m, the torsional spring rate of the pendulum system could be obtained as:

Similarly, when the electric comb voltage ranged from 0 to 100 V the measured displacement voltage straight line of the torsional pendulum fits the same straight line and the fitting slope was1.7 ×10−3(μm V−2)with a linear goodness of fit of 0.9999.It was suggested that torsional pendulum had a good linear ability with small force of thrusters.

It is needed to acquire the dynamic characteristics and parameters of the torsional pendulum system when measuring the impulsive external force.According to equation (7),the oscillation of the torsional pendulum system satisfied the characteristic of exponential attenuation [28], in which the attenuation coefficient was related to the damping rate and the moment of inertia of the pendulum.Therefore, the relationship between the damping coefficient and the moment of inertia could be obtained by recording the change of the voltage signal corresponding to the displacement with time during the decay of the free oscillation of the torsion pendulum.The oscillation attenuation curve of the torsional pendulum system and the fitting line of an exponential model of the envelope line are shown in figure 9.

As shown in figure 9, the horizontal axis was time and the vertical axis was the absolute displacement of the displacement sensor, and the black line was the displacement value of the corresponding point of the torsional pendulum,and the amplitude decreased gradually to the center with time.The blue line was the upper envelope of the displacement oscillation, the red line was the exponential model fitting of

the envelope with the fitting equationy=A1·exp ( −x/t1)+y0.The results showed that the goodness of fit was over 0.9999, indicating that thecurve fitted by the index was relatively accurate.According to the fitting equation,the termt1=119.63 ±0.31,and the damping termcan be obtained according to equation (7) asconsequently the damping ratio could be obtained as:

Figure 10 shows the FFT used to determine the natural frequency of the torsional pendulum oscillation.

Figure 11.The feedback signal versus time for large spans of applied voltage on the electrostatic comb.

As shown in figure 10, the frequency of the damping oscillation of the torsional pendulum was 0.41 Hz, and the full width at half maximum (FWHM) was 0.014 Hz.According to equations(6)and(14),the natural frequency of the system could be obtained asfn=0.41 Hz.The natural frequency of the pendulum system was obtained from the FFT of the oscillation, and the natural frequency of the oscillation was used to determine the moment of inertia according to the equation [28, 35]:

Consequently the moment of inertiaJcould be obtained according to equation (15) as 0.048 kg m .2

3.2.Closed-loop measurement

The closed-loop control method can be used to measure the static force, and its advantage is that it can reduce the influence of external factors such as cables.In the closed-loop mode,the feedback force which is balanced with the torque of external force is applied to the torsional pendulum to maintain the pendulum at a specific position(as shown in figure 3).At the moment when the external force changes, the feedbackcontrol coil applies a balanced force through the PID control circuit,and finally realizes the balance with the external force again.In this experiment, the input signal of the PID control loop was the position signal of the displacement sensor, and the output signal was the current on the feedback coil in figure 1.Therefore,the corresponding variable of the external force measurement was transformed from the position signal to the voltage signal on the feedback device.

Figure 11 exhibits the feedback voltage of the coil when the voltage on the electrostatic comb pair increased from 0 to 450 V with the increment of 50 V and then returned to 0 V.The corresponding thrust of each static thrust step was noted and the enlarged image showed the raw data of electrostatic comb voltage from 0 to 50 V.

Figure 12.The feedback signal versus time for small spans of applied voltage on the electrostatic comb.

Figure 13.The linear fit of the feedback signal versus square of voltage from 0 to 400 V.

Figure 14.Background noise spectrum density of the system in open-loop and closed-loop mode.

Figure 15.The μCAT ignition test image(left)and the displacement signal of pendulum versus time (right).

As shown in figure 11, the electrostatic comb was maintained at the each voltage step for 30 s,corresponding to the electrostatic force range from 0 to 171.1μN previously measured by the electronic balance in figure 4.From the raw data in figure 11, it could be suggested that the feedback voltage signal changed accordingly when the external force increased or decreased, and the electromagnetic force generated by the feedback coil made an equilibrium with the external force torque consequently and the corresponding feedback voltage recorded was about 1.5–8.5 V.In addition,it was suggested from the data line that when the external force changed, there were almost no sinusoidal oscillations produced by the feedback signal as similar as those shown in figure 6.On the contrary, the feedback signal reached the balance position relatively quickly.This was because the PID control circuit has the ability of rapid response, and its differential control branch (as shown in figure 1) could make a certain prediction of the input signal, so as to control the torsional pendulum under the combined effect of PID circuit and made it return to the equilibrium position as soon as possible.

In figure 11,the enlarged image is the raw data line with time around the moment when the voltage of the electrostatic comb increased from 0 to 50 V, which produced an obvious step of the feedback voltage signal at the time when the force was applied.Similarly, when the electrostatic comb voltage ranged from 0 to 100 V and increased at 10 V intervals, the feedback voltage was measured under several relatively smaller external forces.The relationship between the feedback signal and time is shown in figure 12.

As shown in figure 12, the gray line is the raw data and the red line is the filtered data by Savitzky–Golay filter.When the electrostatic comb voltage increased from 0 to 100 V,the feedback voltage signal ranged from 1.1 to 1.5 V approximately.The enlarged image is the corresponding data line when the voltage of electrostatic comb increased from 0 to 30 V.Although the raw data corresponding to the gray line had some signal noise, the filtered signal line still clearly distinguished the feedback voltage corresponding to the external force of submicron Newton magnitude.According to the previous calibration results of external force, when the voltage of electrostatic comb increased from 0 to 10 V, the corresponding increase of equivalent external force was 0.1μN, and the data curve after filtering showed that the voltage signal increased distinguishably, and the amplitude was approximately 5 mV.

In order to evaluate the linearity of deflection displacement,the results of a linear fit of the displacement versus the square of the voltage are shown in figure 13.

As shown in figure 13,the red line is the fitting line of the feedback voltage signal and square of applied voltage when the voltage of the electrostatic comb ranged from 0 to 400 V.The fitting equation is:

The linear goodness of fitR2of the slope value was greater than 0.9999 with the standard error of the slopeδ=1.8 ×10−8,and it was suggested that the feedback voltage had good linearity with external force compared to that of the open-loop mode.According to equations (10) and(15), the relationship between the displacement and the applied force was obtained by eliminating the square term of voltage:

Similarly, when the electric comb voltage was 0–100 V and the interval was 10 V, the measured feedback signal straight line of the torsional pendulum fit the straight line and the fitting relationship had a linear goodness of fit over than 0.9999.It was suggested that torsional pendulum had a good linear ability with the small force of thrusters at the closedloop mode.From the relationship between the fitted force and the feedback voltage,the error was found to be approximately 0.9%in the closed-loop.Considering the effect caused by the procedure of the calibration of the electrostatic comb,and the error of open-loop measurement was approximately 5.3%.

3.3.Background noise analysis

Plasma micro thrusters of sub-micro-Newton to hundreds of micro-Newton thrust usually exhibit low thrust noise,thus the background noise of the thrust measurement system should be low enough to better reflect the parameters and characteristics of the micro thrusters.In this experiment,the output signal of the thrust stand was measured with no external disturbance in the open-loop mode and closed-loop mode respectively after a period of resting until stabilization,and the signal was converted to the equivalent external force perpendicular to the torsional pendulum (Fexat the thruster mounting position in figure 2) by equations (12) and (16), respectively.The background noise spectrum of the system was obtained by analyzing the power spectral density of the signal, as shown in figure 14.

As shown in figure 14, the gray curve is the background noise power spectral density of the system measured in openloop mode,and the red curve is the noise power spectral density in closed-loop mode.In the open-loop mode, the noise power spectral density had a peak near 0.41 Hz with an amplitude of about 0.4which was caused by the inherent oscillation frequency of the torsional system itself,and if the damping method, such as electromagnetic vortex damping, was applied,the background noise generated by the inherent frequency oscillation would be greatly reduced.In addition, this inherent frequency was known and digital filtering methods could be used to reduce the noise here, minimizing interference with the thrust measurements of the plasma micro thrusters.In other frequency bands with frequencies greater than 0.01 Hz, the frequency spectral density of the background noise in open-loop mode was less than 0.1in general, with a roughly trend of decreasing with increasing frequency.

The power spectral density curve shown in the red curve in figure 14 shows that the noise power spectral density in the closed-loop mode generally tended to decrease with the increase of frequency.When the frequency was less than 0.01 Hz,the background noise in closed-loop mode and openloop mode was approximately the same or in one order of magnitude, while with the gradual increase of frequency, the background noise in closed-loop mode decreased significantly, and the amplitude of the noise spectrum in the interval greater than 0.01 Hz was basically less than 0.1and the amplitude in the frequency zone greater than 0.1 Hz was greatly reduced compared with that in the open-loop mode, and the difference was generally about one order of magnitude.It is worth noting that in the closedloop mode there was no peak near the intrinsic frequency of the system as described above, which indicated that the feedback loop could control the motion of the torsional pendulum and keep it as stable as possible in the equilibrium position well.

3.4.Dynamic evaluation of a μCAT

Micro cathode arc thruster (μCAT) is a kind of pulsed miniature plasma thruster based on vacuum arc ablation [9].The main principle is that the vacuum arc is formed between the cathode and anode under the applied electric field and the conductive film, and the metal vapor from the arc ablation is accelerated and ejected out of the thruster to generate thrust under the coupling effect of heat energy and electromagnetic field.Usually μCAT operates in the pulsed mode, and the pulse bit of the thruster is approximately in the magnitude of 0.01–10μN s depending on the size and discharge energy of the thruster,cathode and anode material,discharge parameters and magnetic field characteristics.

In this experiment, the dynamic response of the thrust measurement system operated on the pulsed discharge mode of the μCAT with titanium anode was obtained, and the μCAT ignition test image taken with long exposure and the displacement signal output from the sensor versus time is shown in figure 15.

As shown in figure 15, the μCAT was operating in pulsed mode injecting plasma at the exit,and the process was recorded where the gray curve represented for the raw data and the red curve represented for Savitzky–Golay filter processed data.The μCAT ignited once at the moment of 4.5 s as shown in figure 15, at that time a spike of signal interference was generated due to the transient high current of the thrusters,and then the torsional pendulum started to oscillate.The maximum amplitudeA2maxof 0.56 μm was obtained from the processed results, consequently the impulse value was calculated from equation (5) as 0.07μN s.The error in impulse bit was found to be 21.6% according to the raw data due to the limit of the data recorder, and 3.4% with respect to theJand other parameters.Because of the weak magnetic field of the μCAT in the experiment, the impulse of the μCAT was relatively small and only the magnitude can be evaluated.In the follow-up work the measurement system using more precise displacement sensor will be proposed to make it more suitable for sub-micro-Newton plasma thrusters.

4.Summary and conclusion

In this work, a device for measuring the thrust of plasma micro thrusters with sub-micro-Newton to hundreds micro-Newtons was proposed, and the structure of the device and the measurement principles in open-loop and closed-loop modes were introduced respectively, and a capacitive sensor was used to measure the displacement to reflect the oscillatory motion of the torsional pendulum and the feedback voltage was obtained to indirectly measure the equilibrium force.An electrostatic weak force generation device was used to simulate the external force generated at the thruster mounting position, and the linear fitting relationship between the external force and the output signal of the thrust measurement system could be obtained by the measurement results of the electronic balance previously, and it was suggested that the thrust measurement system had good fitting linearity in the range of 0–200μN approximately, with the ability to resolve external force variations of 0.1μN.Based on the relationship between the angular deflection of the torsional pendulum and the applied force, the torsional spring rate of the system was obtained, and the damping and inertia parameters were obtained consequently from the pendulum decay oscillation.In addition,the analysis results of the background noise suggested that the thrust measurement system was able to achieve a noise level of 0.1/μN Hz or less in open-loop and closed-loop modes in the frequency range from 10 mHz to 5 Hz approximately, which was advantageous for plasma micro thruster parameters and characteristics measurements.The actual micro cathode arc thruster impulse discharge test results showed that the thruster could produce impulses of 0.1μN s, which could be distinguished by the thrust measurement system.

Acknowledgments

This work was partly supported by the Shanghai Engineering Research Center of Space Engine (No.17DZ2280800).