High precision micro-impulse measurements for micro-thrusters based on torsional pendulum and sympathetic resonance techniques Daixian Zhang, Jianjun Wu, Rui Zhang, Hua Zhang, and Zhen He Citation: Review of Scientific Instruments 84, 125113 (2013); doi: 10.1063/1.4850615 View online: http://dx.doi.org/10.1063/1.4850615 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/84/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Volume and surface propellant heating in an electrothermal radio-frequency plasma micro-thruster Appl. Phys. Lett. 105, 054102 (2014); 10.1063/1.4892656 Direct measurement of neutral gas heating in a radio-frequency electrothermal plasma micro-thruster Appl. Phys. Lett. 103, 074101 (2013); 10.1063/1.4818657 A high sensitivity momentum flux measuring instrument for plasma thruster exhausts and diffusive plasmas Rev. Sci. Instrum. 80, 053509 (2009); 10.1063/1.3142477 Development of a two-dimensional dual pendulum thrust stand for Hall thrusters Rev. Sci. Instrum. 78, 115108 (2007); 10.1063/1.2815336 A high-precision five-degree-of-freedom measurement system based on laser collimator and interferometry techniques Rev. Sci. Instrum. 78, 095105 (2007); 10.1063/1.2786272
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REVIEW OF SCIENTIFIC INSTRUMENTS 84, 125113 (2013)
High precision micro-impulse measurements for micro-thrusters based on torsional pendulum and sympathetic resonance techniques Daixian Zhang,a) Jianjun Wu, Rui Zhang, Hua Zhang, and Zhen He College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China
(Received 27 June 2013; accepted 4 December 2013; published online 27 December 2013) A sympathetic resonance theory is analyzed and applied in a newly developed torsional pendulum to measure the micro-impulse produced by a μN s-class ablative pulsed plasma thruster. According to theoretical analysis on the dynamical behaviors of a torsional pendulum, the resonance amplification effect of micro-signals is presented. In addition, a new micro-impulse measurement method based on sympathetic resonance theory is proposed as an improvement of the original single pulse measurement method. In contrast with the single pulse measurement method, the advantages of sympathetic resonance method are significant. First, because of the magnification of vibration signals due to resonance processes, measurement precision for the sympathetic resonance method becomes higher especially in reducing reading error. With an increase in peak number, the relative errors induced by readout of voltage signals decrease to approximately ±1.9% for the sympathetic resonance mode, whereas the relative error in single pulse mode is estimated as ±13.4%. Besides, by using the resonance amplification effect the sympathetic resonance method makes it possible to measure an extremely low-impulse beyond the resolution of a thrust stand without redesigning or purchasing a new one. Moreover, because of the simple operational principle and structure the sympathetic resonance method is much more convenient and inexpensive to be implemented than other high-precision methods. Finally, the sympathetic resonance measurement method can also be applied in other thrust stands to improve further the ability to measure the low-impulse bits. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4850615] I. INTRODUCTION
The measurement of micro-impulses has extensive applications in aerospace and other fields. The impulse bits produced by micro-thrusters applied in spacecrafts may be within the wide ranges of several nano-N s to tens of milli-N s. These various impulse bits are measured in ground experiments for the development of thrusters. Although it is easy to test milliN s or N s class impulse bits, there is an increasing demand in the high precision measurement of impulse bits on the order of several nano-N s or micro-N s when micro-thrusters are developed for attitude and position control, drag compensation, and other space mission applications. For example, microimpulse bits produced by a pulsed plasma thruster is typically several hundred micro-N s.1 A pulsed plasma thruster, which was applied in the preliminary prototype developed for the Chinese TEPO space mission, produced an average impulse bit of about 58.4 μN s as tested by a thrust stand.2 In the NASA and ESA’s Laser Interferometer Space Antenna (LISA)3 for detecting gravitational waves to confirm the Einstein’s theory of General Relativity, micro-N level thrusters are needed to achieve the mission goals. Besides, the NASA ST7 mission calls for thrusters capable of delivering thrust in the 2–20 μN range, with a resolution of 0.1 μN.4 The requirements of high precision measurement of low-impulse bits have been put forward to develop low-impulse or lowthrust thrusters. a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0034-6748/2013/84(12)/125113/11/$30.00
Thrust stand designs for impulse bit measurements have been discussed in the past three decades. A variety of thrust stand designs have been used for thrust measurements, including the hanging pendulum,5–7 inverted pendulum,8–10 and torsional pendulum.11, 12 The history of the torsional pendulum can be traced back to the year 1798 when the universal Newtonian gravitational constant was obtained by S. H. Cavendish using his ingeniously constructed torsion balance.13 The torsion balance transforms micro-displacements into significant quantities that may be observed easily. The micro-displacement is then obtained from the relationship between minor displacements and the significant quantities. Because the direction of the thrust in a torsion-type thrust stand is vertical to Earth’s gravitational acceleration, measurements of impulse bits produced by a thruster can be independent of the thruster mass. Because of its high sensitivity, the statically calibrated torsional pendulum was constructed by Phipps et al.14 to test the nN s-class impulse bits produced by a laser ablation microthruster. The pendulum in the early work of Phipps et al.14 could be separated into a torsional thread (fused silica fiber, diameter: 78 μm), fiber vise, bushing, and an oil cup. The motion of pendulum was recorded by video camera and was calibrated using a gravitational method. A torsional pendulum supported by flexural pivots was applied by Koizumi et al.12 to accurately measure thrust produced by a liquid propellant pulsed plasma thruster (LP-PPT) and a diode laser ablation micro-thruster. The impulse of the LP-PPT ranged from 20 to 80 μN s. The diode laser ablation thruster produced a much lower impulse range of 1–10 μN s for about 1 s. However,
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it was found that the mechanical noise induced from background vibrations becomes a crucial problem for precise estimate of thrust, particularly in low-impulse measurements. To measure μN s-class impulses of a laser-assisted pulsed plasma thruster, a torsional pendulum was developed by Horisawa et al.15 and utilized for thrust performance measurements. The torsional pendulum consisted of a balance, a Csection pivot, a displacement sensor, and a counterweight. Calibration of the torsional pendulum was conducted with known impulses using arbitrary impacts of an aluminum rod. From the velocities of the rod before and after the impacts on the balance measured from images acquired by a high-speed camera, momentum changes or impulses could be obtained. However, calibration was conducted not in a vacuum chamber but in a calibration room. In addition, the damping effect of the atmosphere and errors induced by the non-elastic effects were not considered. In our recent work at the National University of Defense Technology,16–18 the μN s-class ablative pulsed plasma thruster is investigated including characteristics of films deposited by the thruster plume, contamination arising from the exhaust plume, and thrust performance. To improve the deposition characteristics of thruster plume and thrust performance, the pulsed plasma thruster has been redesigned and developed. As the most fundamental measurement made on a pulsed plasma thruster, the assessment of thrust performance is vital. In this work, a new thrust stand based on torsional pendulum techniques is developed to measure thrust performance, including a high precision electromagnetic calibration method and an optical measurement method for microdisplacement in a torsional pendulum. In addition, a newly developed thrust measurement method based on the theory of sympathetic resonance is proposed to improve the accuracy in micro-impulse bit measurements. When the impulse produced by a micro-thruster becomes extremely small, the effective signal will become too small to read out and may be buried by noise signals. The crucial problem induced by noise signals in low-impulse measurements has been illustrated in the work of Koizumi et al.12 Although there are more accurate thrust stands to measure the micro-impulse bit, the thrust stands must be redesigned or purchased. A potential and convenient method to overcome this problem is to amplify the micro-angular signals to significant quantities just like the innovation presented in Cavendish’s experiments. Herein, the sympathetic resonance method is proposed to enlarge the minor vibration signals of a torsional pendulum without changing any parameters of the original thrust stand. The sympathetic vibration in the thrust stand can be conducted when the period of an impulse is adapted to be equal with the vibration period of the pendulum in the thrust stand. As the sympathetic vibration occurs, the swing of the pendulum becomes more and more intense and the corresponding angle signals will be gradually more significant and easier to measure. It becomes convenient to test micro-impulse bits with a much higher accuracy for a certain thrust stand. The dynamic analysis of a torsional pendulum with sympathetic vibration has been conducted. The application of sympathetic vibration theory in thrust measurements for a pulsed plasma thruster has been achieved.
Rev. Sci. Instrum. 84, 125113 (2013)
II. ANALYSIS
A detailed dynamic analysis is necessary to predict the characteristics of a thrust stand with a micro-thruster. For a torsional pendulum rotating around a torsional bar, there are three kinds of moments: the torsional moment M(t) for driving the pendulum, the damping moment Td , and the counteractive moment Tr for relapsing. They can be, respectively, expressed as M(t) = F(t)R, Td = −γ dθ /dt, and Tr = −kθ , where F(t) is the driving force to the pendulum; R is effective length of force arm; θ is the rotational angular displacement of the torsional bar; γ is the coefficient of damping moment; and k is the stiffness coefficient of the pendulum. Together with the definitions of the above three moments, the dynamical equation for describing the vibration of pendulum can be expressed as dθ d 2θ + ω2 θ = F (t)R/J, + 2β dt 2 dt
(1)
where 2β = γ /J, ω2 = k/J, J is the moment of inertia. The initial condition of Eq. (1) is given as θ (0) = θ0
(2)
dθ (t) |t=0 = θ˙0 . dt
(3)
and
Considering that the damping effect is usually much weaker than the effect of a counteractive moment for relapsing of torsion balance, it is assumed that β < ω. The torsional angle can be expressed as e−βt ˙ θ0 sin ω0 t + ωθ0 sin (ω0 t + ϕ) ω0 t R + F (τ ) e−β(t−τ ) sin [ω0 (t − τ )] dτ, (4) J ω0 0 where ω0 = ω2 − β 2 , ϕ = arctan (ω0 /β). When θ˙0 = θ0 = 0, Eq. (4) is simplified to t R F (τ ) e−β(t−τ ) sin [ω0 (t − τ )] dτ. (5) θ (t) = J ω0 0 θ (t) =
A. Single pulse mode
When a torsional pendulum is impacted by a single pulsed thrust of very short duration, the time response characteristic can be analyzed. The pulsed thrust produced by the thruster can be regarded as a single pulsed force with amplitude of F0 and duration of τ p , i.e., ⎧ t τp It is assumed that the duration of a single pulsed thrust is much shorter than a quarter of the period of the torsional oscillation, i.e., τ p < T/4. When 0 ≤ t ≤ τ p , the torsional
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angle can be expressed as
ω −βt F0 R (ω 1 − θ (t) = e sin t + ϕ) . 0 J ω2 ω0
(6)
When t > τ p , the single pulsed thrust disappears and the torsional angle can be calculated as F0 R −β(t−τp ) θ (t) = {e sin[ω0 (t − τp ) + ϕ] J ωω0 −e−βt sin(ω0 t + ϕ)}.
where N = 1, 2, 3, . . . , ∞. When (N − 1)T ≤ t ≤ (N − 1)T + τ p , together with Eqs. (4) and (5) the torsional angle can be calculated as
(7)
Obviously, the period of torsional oscillation of the pendulum is T = 2π/ω0 .
multi-pulsed thrusts are analyzed. Herein, the driving multipulsed thrust is regarded as a periodic force with a period of T as follows: F0 (N − 1)T ≤ t ≤ (N − 1)T + τp , F (t) = 0 (N − 1)T + τp < t ≤ N T
θ (t) =
−e−βτp sin[ω0 (t − nT ) + ϕ]}
(8)
+
Equations (6) and (7) demonstrate that the first peak torsional angle can be achieved in following: sin(ω0 τp ) 1 arctan . tp = ω0 cos(ω0 τp ) − e−βτp
(9)
The time tp is called the first peak time hereafter. The corresponding first peak torsional angle can be obtained as θp =
F0 Re−β(tp −τp ) sin(ω0 τp ) . J ω2 sin(ω0 tp )
(15)
When (N − 1)T + τ p ≤ t ≤ NT, the torsional angle can be determined as N−1 F0 Re−β(t−τp ) βnT e {sin[ω0 (t − τp − nT ) J ωω0 n=0
θ (t) =
(10)
+ϕ] − e−βτp sin[ω0 (t − nT ) + ϕ]}.
(16)
The Nth peak time tp (N) can be obtained as tp (N ) =
(11)
sin(ω0 τp ) 1 arctan + (N − 1)T . ω0 cos(ω0 τp ) − e−βτp
(17)
The corresponding peak torsional angle can be calculated
The first peak torsional angle θ p can be simplified to F0 Rτp exp[−β(ϕ/ω0 − τp )] Jω R ≈ Ib exp(−βϕ/ω0 ). (12) Jω The relationship between the first peak torsional angle θ p and the impulse bit Ib can be expressed as θp =
θp = φIb ,
F0 R {sin ϕ − e−β[t−(N−1)T ] J ωω0
× sin[ω0 (t − (N − 1)T ) + ϕ]}.
Assuming that the duration τ p is as short as several μs and is much shorter than the period T, the first peak time tp in Eq. (9) can be simplified to tp = ϕ/ω0 .
N−2 F0 Re−β(t−τp ) βnT e {sin[ω0 (t − τp − nT ) + ϕ] J ωω0 n=0
(13)
as
×
R exp(−βϕ/ω0 ). (14) Jω The coefficient φ is regarded as the conversion ratio between θ p and Ib . For a certain thrust stand, the conversion ratio φ can be recognized as a constant number that can be obtained according to calibration processes. The peak torsional angle of the torsional pendulum is in direct proportion to the impulse bit Ib . After obtaining the conversion ratio φ from calibration processes and the torsional angular signals θ p from the sensors in the thrust stand, the impulse bits can be calculated from Eq. (13). B. Sympathetic resonance mode
For the application of sympathetic vibration in microthrust measurements, the dynamical characteristics of sympathetic vibration in a torsional pendulum under the impact of
N
n=1
e(n−1)βT . sin{ω0 [tp (n) − (n − 1)T ]}
(18)
Considering duration τ p is as short as several μs, the peak time tp (N) in Eq. (17) can be simplified to
where the coefficient φ can be expressed as φ=
F0 Re−β(tp (N)−τp ) sin(ω0 τp ) J ω2
θp (N ) =
tp (N ) = ϕ/ω0 + (N − 1)T .
(19)
The peak torsional angle in Eq. (18) can be simplified to θp (N ) =
N F0 Rτp −β(tp (N)−τp ) (n−1)βT e e Jω n=1
≈ Ib
R −βϕ/ω0 1 − e−NβT . e Jω 1 − e−βT
(20)
The peak torsional angle is directly proportional to the impulse bit Ib . The relationship between peak torsional angle θ p (N) and the impulse bit Ib can be expressed as θp (N ) = (N )Ib ,
(21)
where the coefficient (N) is a function of the peak number N. It can be expressed as
(N ) =
e−NβT − 1 R −βϕ/ω0 e−NβT − 1 e = φ . Jω e−βT − 1 e−βT − 1
(22)
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The magnification factor of the angular signals due to sympathetic vibration can be expressed as α(N ) =
e−NβT − 1 . e−βT − 1
(23)
III. RESULTS AND DISCUSSIONS A. Description of the torsional pendulum
A calibrated thrust stand utilizing torsional pendulum techniques is developed and applied to estimate μN s-class impulse bits. A photograph and schematic illustration of the thrust stand is shown in Fig. 1. As shown in Fig. 1(a), the torsional pendulum inside the vacuum tank consists of the following: (1) a solid propellant ablative pulsed plasma thruster with a capacitor of 20 μF; (2) a torsional arm; (3) a Csection torsional bar made of beryllium-bronze; (4) a circular mirror (diameter: 12 mm, gold-plating); (5) He-Ne laser (532 nm, power: < 10 mW); (6) rectangular mirrors (with silver plating); (7) a counterweight; (8) a one-dimensional position sensing detector (PSD) for measuring the microdisplacement of the He-Ne laser spot; (9) an electromagnetic calibration device consisting of a pulsed Ampere force generator inside the vacuum tank and a function generator outside the vacuum tank; and (10) an electrical adapter for the electrical connection between the thruster inside the vacuum tank and the power supply and ignition controller outside the vacuum tank. As shown in Fig. 1(b), the pulsed plasma thruster is set onto the torsional pendulum, and the length between the axis of torsional bar and the thruster is labeled as R, which is regarded as the effective length of the force arm (R = 0.520 ± 0.001 m). The solid propellant Polytetrafluoroethylene (PTFE) is used. The thruster is composed of a single capacitor, parallel plate electrodes, and an ignitor plug. Various voltages (300 − 1500 V) are charged to the electrodes via a capacitor (20 μF). The distance between the electrodes
is 30 mm, and the width and length of the electrodes are 15 and 25 mm, respectively. The ignition energy produced by the ignitor plug is about 0.3 J. A power supply and an ignition controller are fixed outside the vacuum tank. When the command of ignition is exported, the thruster will start to work, and the torsional pendulum will swing around the torsion bar under the pulsed thrust produced by the thruster. The angular displacement of the pendulum is recorded by an optical measurement system, which is composed of (4) a circular mirror, (5) a He-Ne laser, (6) rectangular mirrors, (8) a PSD sensor, the data acquisition device, and the computer as shown in Fig. 1(b). The solid He-Ne laser beam irradiates the circular mirror and eventually the PSD sensor after being reflected several times by mirrors as shown in Fig. 1(b). The length of the optical path between the circular mirror and the PSD is labeled as l(l = 3.250 ± 0.001 m). Obviously, when the angular displacement of the pendulum is θ (t), the deviation angle of the laser beam irradiating the PSD sensitive surface becomes 2θ (t). Thus, the offset of the laser beam spot on the surface of PSD will be 2θ (t)l. The relationship between a torsional angle and the corresponding voltage signal of the PSD can be expressed as U (t) = (4Vmax l/ lP SD )θ (t) = ηθ (t), where lPSD is the effective optical sensitive length of the PSD (lP SD = 30.0 mm), U(t) is the voltage signal produced by the PSD, and Vmax is the potential maximal value of the voltage signal produced by the PSD (Vmax = 3.0 V). The relationship between the peak PSD signals and peak torsional angles can be expressed as Up = ηθp .
(24)
The conversion ratio between torsional angles and voltage signals of the PSD can be regarded as a constant due to the high linearity of the PSD sensor and is expressed as η = 4Vmax l/ lP SD = (1300 ± 43.34) V/rad. Calibration procedures must be conducted to measure impulse bits. An electromagnetic calibration device based on Ampere force is constructed as shown in Figs. 1 and 2. The Ampere force is generated by a pulsed generator consisting of
FIG. 1. Torsional pendulum. (a) Photo. (b) Schematic illustration (top view).
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FIG. 2. Pulsed Ampere force generator. (a) Photo. (b) Schematic illustration.
two separate parts as shown in Fig. 2(a). Part A consists of two parallel permanent magnetic plates (material: NdFeB, labeled as 11) and a bracket made of organic glass. Part B consists of circles of conducting copper wires (labeled as 12) and support made of organic glass as shown in Fig. 2. Besides, a function generator is used to produce a certain current IC for the Ampere force generator in the calibration process as shown in Fig. 1. The Ampere force can be expressed as FA = mBIC LC , determined by the number of turns m, magnetic induction B between parallel permanent magnet plates, and current IC and effective length LC of the conducting copper wires. The pulsed Ampere force produced by the pulsed Ampere force generator should be measured and checked before the application of the calibration device. Assembled with Part B, Part A of the generator can be weighed by an electronic mass balance (sensitivity: 10 μg, maximal mass: 80 g). When the current IC inside part B is adapted by the function generator, the digital reading of the weight of Part A changes because the supporting force from the electronic mass balance will decrease or increase owing to the direction of the Ampere force. Obviously, the variation of the digital reading represents the Ampere force produced by the generator under the current IC . The relationship between the current of the Ampere force generator and the Ampere force can be measured as shown in Fig. 3, and expressed as FA = AAmpere + BAmpere IC ,
(25)
Ampere force FA [μN]
1000
500
0
0
0.05 0.1 0.15 Current of Ampere force generator IC [A]
0.2
FIG. 3. Measured curves for Ampere force versus electric current.
where AAmpere = (0.007 ± 0.158) μN, and BAmpere = (5.73 ± 0.02) × 103 μN/A. Here, the linearity of the curve is sufficient for a calibration process. After obtaining the relationship of Eq. (25), Parts A and B are set onto the torsional arm and the X-Y two-dimensional displacement platform respectively as shown in Fig. 2. The distance between the axis of the torsional bar and the centerline of the Part A is denoted by L in Fig. 2(a) and is tested as (0.290 ± 0.001) m. Considering the difference between effective length R and L, the effective force impacting on the pendulum at the location of the thruster can be calculated as F (t) = FA (t)L/R.
(26)
When the pendulum is propelled by the Ampere force produced by the pulsed Ampere force generator, the pendulum will swing around the torsional bar. Especially when the Ampere force is produced repetitively with a same period as the vibration period of torsional pendulum the sympathetic vibration occurs. B. Measurement in single pulse mode
The impulse bit or single pulse produced by the ablative pulsed plasma thruster can be measured using the torsional pendulum. The calibration is conducted first. Given a certain current IC and duration τ A , the pulsed Ampere force generator produces the corresponding Ampere force and the known impulse bit Ib . As shown in Fig. 4, given a current IC of 197 mA and a duration of 0.1 s and considering the relationship given by Eq. (25), a pulsed Ampere force FA (t) with an amplitude of 1129.95 μN and a duration of 0.1 s can be produced. Together with Eq. (26), the corresponding impulse bit can be expressed as Ib = F(t)τ A = FA (t)τ A L/R = 0.5577 × 112.995 μN s = 63.0 μN s. Under this known impulse bit, the torsional pendulum can be propelled as shown in Fig. 4. In Fig. 5, the first peak PSD signal can be tested and labeled as Up, 0 . Under various impulse bits produced by pulsed Ampere force generator, the corresponding peak PSD signals can be obtained. The relationship between impulse bits and peak PSD signals is Up,0 = A0 + B0 Ib ,
(27)
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0.15
Pulsed Ampere force ends
Up,0 0
0
-0.05
0
2 Time [s]
4
-0.15
(c)
-0.15 0
where A0 = (−3.9 ± 253.0) × 10−5 V, B0 = (2.72 ± 0.34) × 10−4 V/μN s, the unit of Ib is μN s, and the unit of Up, 0 is V. The subscript 0 in Up, 0 indicates that the quantity Up, 0 is obtained in a calibration process. After calibration processes, the measurements of an impulse can be achieved. The time histories of PSD signals under different charged energies in the pulsed plasma thruster are listed in Fig. 6. The first peak PSD signals Up can be read out from the figure. Together with Eq. (27), the impulse bits under different charged energies (EC = 6.4, 12.1, 22.5 J) are obtained as shown in Fig. 7. Five measurements of impulse bits are conducted for every charged energy of capacitors in the experiments. The impulse bits increase with the charged energy of capacitors in the pulsed plasma thruster. The single pulse measurement mode of the thrust stand appears sufficient to measure thrust performance of the micro-thruster. However, when impulse bits decrease with charged energies, the corresponding torsional angles become extremely small and PSD signals become too small to be measured with high accuracy. In fact, when PSD signals decrease to orders of several mV or less, noise signals may partially cover the real PSD signals. As a result, the reading error of the PSD signals becomes more and more significant with a decrease in impulse bits. The detailed analysis of the noise signals in the thrust stands can be found in many past works.12 We do not intend to put emphasis on the analysis of the origin of complex noise signals in a thrust stand, though the noise of thrust stand is not negligible when PSD signals become small. Obviously,
Time [s]
5
10
FIG. 6. Time histories of PSD signals under different working conditions of the thruster. (a) EC = 6.4 J. (b) EC = 12.1 J. (c) EC = 22.5 J.
in single pulse measurement mode, measurement accuracy in the thrust stand will decrease with a decrease in impulse bits that we wish to measure.
C. Measurement in sympathetic resonance mode
According to the above analysis, it is difficult to test lowimpulse bits with high accuracy in single pulse mode. Therefore, higher precision and accuracy of measurement are expected according to the usage of sympathetic vibration in the thrust stand. To evaluate the ability to measure low-impulse bits produced by the thruster, the charged energy of the capacitors in the thruster is set at a low level of 6.4 J with the corresponding impulse bit is about 95 μN s as tested in single pulse mode as shown in Fig. 7. First, the calibration procedure is conducted by using the pulsed Ampere force generator as illustrated in Figs. 1 and 2. The function generator is used to control the Ampere force. Prior to setting the function generator, the period of the torsional pendulum is tested to be approximately 1.310 s. The period of the pulsed Ampere force is set to be the same as the period of the torsional pendulum for the formation of sympathetic vibration. Given the magnitude and duration of current in the pulsed Ampere generator, the certain impulse bits can be produced to propel the torsional arm. Fig. 8 shows two time histories of sympathetic vibrations produced by the pulsed 600
0.05
500
0.04
Impulse Bit Ib [μN⋅s]
Peak PSD signal Up,0 [V]
(b)
0
0
6
FIG. 4. A time history of PSD signal under a pulsed Ampere force.
0.03 0.02 0.01 0
0.15
0.15
-1000
Pulsed Ampere force starts -0.1 -2
-0.15 PSD signal [V]
PSD signal [V]
PSD signal
Ampere force [μN]
1000 τA=0.1 s
0.05
(a)
0
400 300 200 100
0
50 100 Impulse Bit Ib [μN⋅s]
FIG. 5. Peak PSD signals versus impulse bits.
150
0
0
5
10 15 Charged Energy EC [J]
20
25
FIG. 7. The impulse bit versus charged energy.
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0.3
1000
Pulsed Ampere Force
0
0.2
0 PSD Signal
-1000
-0.5
(b)
0.5
Pulsed Ampere Force
1000
0
0
-0.5 -5
PSD Signal 0
5
20
25
0 -0.1 -0.2
-1000 10 15 Time [s]
0.1
30
0
FIG. 8. Sympathetic vibrations produced by pulsed Ampere force generator. (a) Current amplitude: 100 mA, τ A = 0.1 s. (b) Current amplitude: 197 mA, τ A = 0.1 s.
Ampere force generator with different driving currents for the generator. The theoretical value of the magnification factor α(N) can be obtained from Eq. (23). It can be found that when β = 0, α(N) = N. However, the damping effect always exists, i.e., β > 0, therefore, α(N) < N. The damping effect with different β (β = 0, 0.01, 0.1, 0.5 Hz) and T = 1.310 s can be observed in Fig. 9. With an increase in β, the nonlinearity between the magnification factor and the peak number becomes more and more significant. From the calibration process as shown in Fig. 8, the magnification factor α(N) can be measured and is listed in Fig. 9. This indicates that though α(N) < N because of the damping effect, the nonlinearity is insignificant for the thrust stand within N < 20 as listed. When the torsional pendulum is working in a resonance mode, the peak voltage is tested to be 0.1 Hz, for example), the resonance amplification effect will be damped more intensively. Fortunately, the influence of nonlinearity can be overcome effectively by adding more calibration data points when the effect of nonlinearity becomes significant; i.e., though the
5 Time [s]
10
FIG. 10. Sympathetic vibrations produced by the thruster.
calibrated curve is nonlinear, the calibration data can be applied in the impulse measurement. The time histories of PSD signals can be tested for different impulses produced by the pulsed Ampere force generator, and the magnification factors can be demonstrated to be independent of the impulses. For every peak number N, the relationship between impulse bits and peak PSD signals can be obtained and expressed as follows: Up,0 (N ) = AN + BN Ib ,
(28)
where BN is regarded as a function of the peak number N, i.e., BN ≈ ηφα(N), and both BN and AN can be easily obtained from the calibration process. Equation (27) can be regarded as the special case of Eq. (28) when N = 1, and then sympathetic resonance mode degenerates into single pulse mode. Besides, it can be demonstrated that BN ≈ B0 α(N). By adapting the power supply and ignition controller, the period of impulse bits produced by the thruster can be changed to be same with the torsional period of the pendulum. The sympathetic vibration produced by the thruster are measured as shown in Fig. 10. Peak PSD signals Up (N) for different times of sympathetic vibrations can be read out. Note that Up, 0 (N) is the peak PSD signals in the calibration process without the thruster working and Up (N) is the peak PSD signals in follow-up measurement with the thruster working. The corresponding impulse bits can be obtained as shown in Fig. 11 by using the calibrated relationship between impulse bits and peak PSD signals given by Eq. (28). The variation of 150
0.3 Calibration Data Analytical Data
1 2
15
1- β=0 2- β=0.01 3- β=0.1 10 4- β=0.5 (Unit- Hz)
3
5 4 0
0
5
10 15 Peak number N
FIG. 9. Magnification factor versus peak number.
20
Peak PSD Voltage Up(N) [V]
α(N) or Up,0(N)/(ηφIb)
20
Peak PSD Voltage Impulse Bit 0.2
100
0.1
50
0
0
1
2
3
4 5 6 7 Peak Number N
8
9
Impulse Bit Ib [μN⋅s]
(a)
PSD signal [V]
PSD Signal [V]
0.5
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Ampere Force [μN]
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0 10
FIG. 11. Variation of peak PSD voltage and impulse bit with peak number.
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impulse bits with peak number N is obtained within ranges of 88–96 μN s as shown in Fig. 11. Besides, it can be found that an impulse bit of about 91 μN s is obtained when the number of peaks is raised to 9. As shown in Fig. 7, the impulse bits under the charged energy of 6.4 J are obtained with a range of 89.2–108.2 μN s. This indicates that the results in sympathetic resonance mode are in well accordance with the results in single pulse mode. The results obtained by the sympathetic resonance measurement method are reliable, indeed. Compared with single pulse mode, the advantages of sympathetic resonance method lie in following aspects: (1) Higher precision in measuring the low impulse bits is achieved for the sympathetic resonance method, which will be discussed in the section on Uncertainty analysis, and (2) the sympathetic resonance method makes it possible to measure extremely low-impulse bits, which are difficult to test for the single pulse method if the original thrust stand is not improved or if another high precision method is not applied. Besides, because of the simple operational principle and structure the sympathetic resonance method is much more convenient and inexpensive to be implemented than other high precision methods, such as the laser interferometer method. D. Uncertainty analysis
Uncertainties of measurements in single pulse and sympathetic resonance modes are analyzed for thruster with a charged energy of 6.4 J. The possible sources of uncertainty within impulse measurements are listed herein. 1. Uncertainty induced by fluctuations of impulse bits
In the above theoretical analysis, the impulses for every peak number are assumed to be the same. However, the interval and magnitude of the impulses or the fine structure of the thrust curve may vary for every real pulsed thrust. The irregular interval and magnitude of impulse bits lead to fluctuations of impulse bits when the thrust works in resonance mode. Thus, errors of measurements may be induced by fluctuations of the impulse bits. It is reasonable to assume that there is a fluctuation δIb (n) for impulse bit, i.e., Ib (n) = I b (n) + δIb (n), where I b (n) is the mean value of the impulse bit. It is assumed that I b (n) = a · n + δ(n), where a [μN s/pulse] is the speed for a downtrend or uptrend of impulse bits, and δ(n) is the random part of the fluctuations. The relationship between the peak angles and the impulse bits can be obtained as θp (N) = φ[α(N )I b + aN (N + 1)/2 + N n=1 δ(n)]. The measured value of the impulse bit can be obtained as I˜b (N) = θp (N)/(φα(N )). The relative errors of impulse bits induced by fluctuations of impulses bits can be expressed as Ib (N )/Ib (N ) = {aN [N + 1 − 2α(N)]/2 + N n=1 δ(n)}/[α(N )Ib (N )]. For an optimal pulsed plasma thruster, the long-term stability of thrust performance is expected and realizable without a significant tendency of increment or decrement of impulse bits, i.e., a ≈ 0 [μN s/pulse]. Besides, the random fluctuations of impulse bits (δ(n)) are expected to be sufficiently low in contrast with the impulse bit Ib , i.e., δ(n) Ib (n). There-
Rev. Sci. Instrum. 84, 125113 (2013)
fore, for an optimal thruster, the relative errors induced by the fluctuations of impulse bits are negligible, i.e., Ib (N)/Ib (N) ≈ 0. In this article, the variations of impulse bits are observed and it is reasonable to assume that |a| < 0.1 μN s/pulse. The errors induced by random fluctuations δ(n) can be estimated as Ib (N )/Ib (N ) = N n=1 δ(n)/α(N )Ib (N ). Considering the uniformly distributed random fluctuations, the summation of fluctuations δ(n) becomes zero when the peak number N extends to infinity. In addition, the errors induced by random fluctuation δ(n) of an impulse bit can be reduced with the increment of peak number and can be regarded as negligible when peak number N increases to 10. For a peak number N < 10 and an impulse bit of 100 μN s, the relative errors with the exception of the influence of random fluctuations δ(n) can be estimated as |Ib (N)/Ib (N)| ≈ |{aN[N + 1 − 2α(N)]}/[2α(N)Ib (N)]| < 0.5%. Obviously, errors induced by fluctuations of impulse bits are controllable for the sympathetic resonance method. By improving the stability of thrust performance, the errors induced by fluctuations of impulse bits become insignificant. 2. Uncertainty in displacement measurement of the He-Ne laser spot using a PSD sensor
Considering that the coefficient A0 in Eq. (27) can be negligible, Eq. (27) together with Eq. (24) can be changed to θ p = B0 Ib /η. The conversion ratio φ in Eq. (13) can be obtained as φ = B0 /η = 2.1 × 10−7 rad/μN s. The displacement resolution of the PSD sensor, i.e., δlPSD , is
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REVIEW OF SCIENTIFIC INSTRUMENTS 84, 125113 (2013)
High precision micro-impulse measurements for micro-thrusters based on torsional pendulum and sympathetic resonance techniques Daixian Zhang,a) Jianjun Wu, Rui Zhang, Hua Zhang, and Zhen He College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China
(Received 27 June 2013; accepted 4 December 2013; published online 27 December 2013) A sympathetic resonance theory is analyzed and applied in a newly developed torsional pendulum to measure the micro-impulse produced by a μN s-class ablative pulsed plasma thruster. According to theoretical analysis on the dynamical behaviors of a torsional pendulum, the resonance amplification effect of micro-signals is presented. In addition, a new micro-impulse measurement method based on sympathetic resonance theory is proposed as an improvement of the original single pulse measurement method. In contrast with the single pulse measurement method, the advantages of sympathetic resonance method are significant. First, because of the magnification of vibration signals due to resonance processes, measurement precision for the sympathetic resonance method becomes higher especially in reducing reading error. With an increase in peak number, the relative errors induced by readout of voltage signals decrease to approximately ±1.9% for the sympathetic resonance mode, whereas the relative error in single pulse mode is estimated as ±13.4%. Besides, by using the resonance amplification effect the sympathetic resonance method makes it possible to measure an extremely low-impulse beyond the resolution of a thrust stand without redesigning or purchasing a new one. Moreover, because of the simple operational principle and structure the sympathetic resonance method is much more convenient and inexpensive to be implemented than other high-precision methods. Finally, the sympathetic resonance measurement method can also be applied in other thrust stands to improve further the ability to measure the low-impulse bits. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4850615] I. INTRODUCTION
The measurement of micro-impulses has extensive applications in aerospace and other fields. The impulse bits produced by micro-thrusters applied in spacecrafts may be within the wide ranges of several nano-N s to tens of milli-N s. These various impulse bits are measured in ground experiments for the development of thrusters. Although it is easy to test milliN s or N s class impulse bits, there is an increasing demand in the high precision measurement of impulse bits on the order of several nano-N s or micro-N s when micro-thrusters are developed for attitude and position control, drag compensation, and other space mission applications. For example, microimpulse bits produced by a pulsed plasma thruster is typically several hundred micro-N s.1 A pulsed plasma thruster, which was applied in the preliminary prototype developed for the Chinese TEPO space mission, produced an average impulse bit of about 58.4 μN s as tested by a thrust stand.2 In the NASA and ESA’s Laser Interferometer Space Antenna (LISA)3 for detecting gravitational waves to confirm the Einstein’s theory of General Relativity, micro-N level thrusters are needed to achieve the mission goals. Besides, the NASA ST7 mission calls for thrusters capable of delivering thrust in the 2–20 μN range, with a resolution of 0.1 μN.4 The requirements of high precision measurement of low-impulse bits have been put forward to develop low-impulse or lowthrust thrusters. a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0034-6748/2013/84(12)/125113/11/$30.00
Thrust stand designs for impulse bit measurements have been discussed in the past three decades. A variety of thrust stand designs have been used for thrust measurements, including the hanging pendulum,5–7 inverted pendulum,8–10 and torsional pendulum.11, 12 The history of the torsional pendulum can be traced back to the year 1798 when the universal Newtonian gravitational constant was obtained by S. H. Cavendish using his ingeniously constructed torsion balance.13 The torsion balance transforms micro-displacements into significant quantities that may be observed easily. The micro-displacement is then obtained from the relationship between minor displacements and the significant quantities. Because the direction of the thrust in a torsion-type thrust stand is vertical to Earth’s gravitational acceleration, measurements of impulse bits produced by a thruster can be independent of the thruster mass. Because of its high sensitivity, the statically calibrated torsional pendulum was constructed by Phipps et al.14 to test the nN s-class impulse bits produced by a laser ablation microthruster. The pendulum in the early work of Phipps et al.14 could be separated into a torsional thread (fused silica fiber, diameter: 78 μm), fiber vise, bushing, and an oil cup. The motion of pendulum was recorded by video camera and was calibrated using a gravitational method. A torsional pendulum supported by flexural pivots was applied by Koizumi et al.12 to accurately measure thrust produced by a liquid propellant pulsed plasma thruster (LP-PPT) and a diode laser ablation micro-thruster. The impulse of the LP-PPT ranged from 20 to 80 μN s. The diode laser ablation thruster produced a much lower impulse range of 1–10 μN s for about 1 s. However,
84, 125113-1
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it was found that the mechanical noise induced from background vibrations becomes a crucial problem for precise estimate of thrust, particularly in low-impulse measurements. To measure μN s-class impulses of a laser-assisted pulsed plasma thruster, a torsional pendulum was developed by Horisawa et al.15 and utilized for thrust performance measurements. The torsional pendulum consisted of a balance, a Csection pivot, a displacement sensor, and a counterweight. Calibration of the torsional pendulum was conducted with known impulses using arbitrary impacts of an aluminum rod. From the velocities of the rod before and after the impacts on the balance measured from images acquired by a high-speed camera, momentum changes or impulses could be obtained. However, calibration was conducted not in a vacuum chamber but in a calibration room. In addition, the damping effect of the atmosphere and errors induced by the non-elastic effects were not considered. In our recent work at the National University of Defense Technology,16–18 the μN s-class ablative pulsed plasma thruster is investigated including characteristics of films deposited by the thruster plume, contamination arising from the exhaust plume, and thrust performance. To improve the deposition characteristics of thruster plume and thrust performance, the pulsed plasma thruster has been redesigned and developed. As the most fundamental measurement made on a pulsed plasma thruster, the assessment of thrust performance is vital. In this work, a new thrust stand based on torsional pendulum techniques is developed to measure thrust performance, including a high precision electromagnetic calibration method and an optical measurement method for microdisplacement in a torsional pendulum. In addition, a newly developed thrust measurement method based on the theory of sympathetic resonance is proposed to improve the accuracy in micro-impulse bit measurements. When the impulse produced by a micro-thruster becomes extremely small, the effective signal will become too small to read out and may be buried by noise signals. The crucial problem induced by noise signals in low-impulse measurements has been illustrated in the work of Koizumi et al.12 Although there are more accurate thrust stands to measure the micro-impulse bit, the thrust stands must be redesigned or purchased. A potential and convenient method to overcome this problem is to amplify the micro-angular signals to significant quantities just like the innovation presented in Cavendish’s experiments. Herein, the sympathetic resonance method is proposed to enlarge the minor vibration signals of a torsional pendulum without changing any parameters of the original thrust stand. The sympathetic vibration in the thrust stand can be conducted when the period of an impulse is adapted to be equal with the vibration period of the pendulum in the thrust stand. As the sympathetic vibration occurs, the swing of the pendulum becomes more and more intense and the corresponding angle signals will be gradually more significant and easier to measure. It becomes convenient to test micro-impulse bits with a much higher accuracy for a certain thrust stand. The dynamic analysis of a torsional pendulum with sympathetic vibration has been conducted. The application of sympathetic vibration theory in thrust measurements for a pulsed plasma thruster has been achieved.
Rev. Sci. Instrum. 84, 125113 (2013)
II. ANALYSIS
A detailed dynamic analysis is necessary to predict the characteristics of a thrust stand with a micro-thruster. For a torsional pendulum rotating around a torsional bar, there are three kinds of moments: the torsional moment M(t) for driving the pendulum, the damping moment Td , and the counteractive moment Tr for relapsing. They can be, respectively, expressed as M(t) = F(t)R, Td = −γ dθ /dt, and Tr = −kθ , where F(t) is the driving force to the pendulum; R is effective length of force arm; θ is the rotational angular displacement of the torsional bar; γ is the coefficient of damping moment; and k is the stiffness coefficient of the pendulum. Together with the definitions of the above three moments, the dynamical equation for describing the vibration of pendulum can be expressed as dθ d 2θ + ω2 θ = F (t)R/J, + 2β dt 2 dt
(1)
where 2β = γ /J, ω2 = k/J, J is the moment of inertia. The initial condition of Eq. (1) is given as θ (0) = θ0
(2)
dθ (t) |t=0 = θ˙0 . dt
(3)
and
Considering that the damping effect is usually much weaker than the effect of a counteractive moment for relapsing of torsion balance, it is assumed that β < ω. The torsional angle can be expressed as e−βt ˙ θ0 sin ω0 t + ωθ0 sin (ω0 t + ϕ) ω0 t R + F (τ ) e−β(t−τ ) sin [ω0 (t − τ )] dτ, (4) J ω0 0 where ω0 = ω2 − β 2 , ϕ = arctan (ω0 /β). When θ˙0 = θ0 = 0, Eq. (4) is simplified to t R F (τ ) e−β(t−τ ) sin [ω0 (t − τ )] dτ. (5) θ (t) = J ω0 0 θ (t) =
A. Single pulse mode
When a torsional pendulum is impacted by a single pulsed thrust of very short duration, the time response characteristic can be analyzed. The pulsed thrust produced by the thruster can be regarded as a single pulsed force with amplitude of F0 and duration of τ p , i.e., ⎧ t τp It is assumed that the duration of a single pulsed thrust is much shorter than a quarter of the period of the torsional oscillation, i.e., τ p < T/4. When 0 ≤ t ≤ τ p , the torsional
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angle can be expressed as
ω −βt F0 R (ω 1 − θ (t) = e sin t + ϕ) . 0 J ω2 ω0
(6)
When t > τ p , the single pulsed thrust disappears and the torsional angle can be calculated as F0 R −β(t−τp ) θ (t) = {e sin[ω0 (t − τp ) + ϕ] J ωω0 −e−βt sin(ω0 t + ϕ)}.
where N = 1, 2, 3, . . . , ∞. When (N − 1)T ≤ t ≤ (N − 1)T + τ p , together with Eqs. (4) and (5) the torsional angle can be calculated as
(7)
Obviously, the period of torsional oscillation of the pendulum is T = 2π/ω0 .
multi-pulsed thrusts are analyzed. Herein, the driving multipulsed thrust is regarded as a periodic force with a period of T as follows: F0 (N − 1)T ≤ t ≤ (N − 1)T + τp , F (t) = 0 (N − 1)T + τp < t ≤ N T
θ (t) =
−e−βτp sin[ω0 (t − nT ) + ϕ]}
(8)
+
Equations (6) and (7) demonstrate that the first peak torsional angle can be achieved in following: sin(ω0 τp ) 1 arctan . tp = ω0 cos(ω0 τp ) − e−βτp
(9)
The time tp is called the first peak time hereafter. The corresponding first peak torsional angle can be obtained as θp =
F0 Re−β(tp −τp ) sin(ω0 τp ) . J ω2 sin(ω0 tp )
(15)
When (N − 1)T + τ p ≤ t ≤ NT, the torsional angle can be determined as N−1 F0 Re−β(t−τp ) βnT e {sin[ω0 (t − τp − nT ) J ωω0 n=0
θ (t) =
(10)
+ϕ] − e−βτp sin[ω0 (t − nT ) + ϕ]}.
(16)
The Nth peak time tp (N) can be obtained as tp (N ) =
(11)
sin(ω0 τp ) 1 arctan + (N − 1)T . ω0 cos(ω0 τp ) − e−βτp
(17)
The corresponding peak torsional angle can be calculated
The first peak torsional angle θ p can be simplified to F0 Rτp exp[−β(ϕ/ω0 − τp )] Jω R ≈ Ib exp(−βϕ/ω0 ). (12) Jω The relationship between the first peak torsional angle θ p and the impulse bit Ib can be expressed as θp =
θp = φIb ,
F0 R {sin ϕ − e−β[t−(N−1)T ] J ωω0
× sin[ω0 (t − (N − 1)T ) + ϕ]}.
Assuming that the duration τ p is as short as several μs and is much shorter than the period T, the first peak time tp in Eq. (9) can be simplified to tp = ϕ/ω0 .
N−2 F0 Re−β(t−τp ) βnT e {sin[ω0 (t − τp − nT ) + ϕ] J ωω0 n=0
(13)
as
×
R exp(−βϕ/ω0 ). (14) Jω The coefficient φ is regarded as the conversion ratio between θ p and Ib . For a certain thrust stand, the conversion ratio φ can be recognized as a constant number that can be obtained according to calibration processes. The peak torsional angle of the torsional pendulum is in direct proportion to the impulse bit Ib . After obtaining the conversion ratio φ from calibration processes and the torsional angular signals θ p from the sensors in the thrust stand, the impulse bits can be calculated from Eq. (13). B. Sympathetic resonance mode
For the application of sympathetic vibration in microthrust measurements, the dynamical characteristics of sympathetic vibration in a torsional pendulum under the impact of
N
n=1
e(n−1)βT . sin{ω0 [tp (n) − (n − 1)T ]}
(18)
Considering duration τ p is as short as several μs, the peak time tp (N) in Eq. (17) can be simplified to
where the coefficient φ can be expressed as φ=
F0 Re−β(tp (N)−τp ) sin(ω0 τp ) J ω2
θp (N ) =
tp (N ) = ϕ/ω0 + (N − 1)T .
(19)
The peak torsional angle in Eq. (18) can be simplified to θp (N ) =
N F0 Rτp −β(tp (N)−τp ) (n−1)βT e e Jω n=1
≈ Ib
R −βϕ/ω0 1 − e−NβT . e Jω 1 − e−βT
(20)
The peak torsional angle is directly proportional to the impulse bit Ib . The relationship between peak torsional angle θ p (N) and the impulse bit Ib can be expressed as θp (N ) = (N )Ib ,
(21)
where the coefficient (N) is a function of the peak number N. It can be expressed as
(N ) =
e−NβT − 1 R −βϕ/ω0 e−NβT − 1 e = φ . Jω e−βT − 1 e−βT − 1
(22)
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The magnification factor of the angular signals due to sympathetic vibration can be expressed as α(N ) =
e−NβT − 1 . e−βT − 1
(23)
III. RESULTS AND DISCUSSIONS A. Description of the torsional pendulum
A calibrated thrust stand utilizing torsional pendulum techniques is developed and applied to estimate μN s-class impulse bits. A photograph and schematic illustration of the thrust stand is shown in Fig. 1. As shown in Fig. 1(a), the torsional pendulum inside the vacuum tank consists of the following: (1) a solid propellant ablative pulsed plasma thruster with a capacitor of 20 μF; (2) a torsional arm; (3) a Csection torsional bar made of beryllium-bronze; (4) a circular mirror (diameter: 12 mm, gold-plating); (5) He-Ne laser (532 nm, power: < 10 mW); (6) rectangular mirrors (with silver plating); (7) a counterweight; (8) a one-dimensional position sensing detector (PSD) for measuring the microdisplacement of the He-Ne laser spot; (9) an electromagnetic calibration device consisting of a pulsed Ampere force generator inside the vacuum tank and a function generator outside the vacuum tank; and (10) an electrical adapter for the electrical connection between the thruster inside the vacuum tank and the power supply and ignition controller outside the vacuum tank. As shown in Fig. 1(b), the pulsed plasma thruster is set onto the torsional pendulum, and the length between the axis of torsional bar and the thruster is labeled as R, which is regarded as the effective length of the force arm (R = 0.520 ± 0.001 m). The solid propellant Polytetrafluoroethylene (PTFE) is used. The thruster is composed of a single capacitor, parallel plate electrodes, and an ignitor plug. Various voltages (300 − 1500 V) are charged to the electrodes via a capacitor (20 μF). The distance between the electrodes
is 30 mm, and the width and length of the electrodes are 15 and 25 mm, respectively. The ignition energy produced by the ignitor plug is about 0.3 J. A power supply and an ignition controller are fixed outside the vacuum tank. When the command of ignition is exported, the thruster will start to work, and the torsional pendulum will swing around the torsion bar under the pulsed thrust produced by the thruster. The angular displacement of the pendulum is recorded by an optical measurement system, which is composed of (4) a circular mirror, (5) a He-Ne laser, (6) rectangular mirrors, (8) a PSD sensor, the data acquisition device, and the computer as shown in Fig. 1(b). The solid He-Ne laser beam irradiates the circular mirror and eventually the PSD sensor after being reflected several times by mirrors as shown in Fig. 1(b). The length of the optical path between the circular mirror and the PSD is labeled as l(l = 3.250 ± 0.001 m). Obviously, when the angular displacement of the pendulum is θ (t), the deviation angle of the laser beam irradiating the PSD sensitive surface becomes 2θ (t). Thus, the offset of the laser beam spot on the surface of PSD will be 2θ (t)l. The relationship between a torsional angle and the corresponding voltage signal of the PSD can be expressed as U (t) = (4Vmax l/ lP SD )θ (t) = ηθ (t), where lPSD is the effective optical sensitive length of the PSD (lP SD = 30.0 mm), U(t) is the voltage signal produced by the PSD, and Vmax is the potential maximal value of the voltage signal produced by the PSD (Vmax = 3.0 V). The relationship between the peak PSD signals and peak torsional angles can be expressed as Up = ηθp .
(24)
The conversion ratio between torsional angles and voltage signals of the PSD can be regarded as a constant due to the high linearity of the PSD sensor and is expressed as η = 4Vmax l/ lP SD = (1300 ± 43.34) V/rad. Calibration procedures must be conducted to measure impulse bits. An electromagnetic calibration device based on Ampere force is constructed as shown in Figs. 1 and 2. The Ampere force is generated by a pulsed generator consisting of
FIG. 1. Torsional pendulum. (a) Photo. (b) Schematic illustration (top view).
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FIG. 2. Pulsed Ampere force generator. (a) Photo. (b) Schematic illustration.
two separate parts as shown in Fig. 2(a). Part A consists of two parallel permanent magnetic plates (material: NdFeB, labeled as 11) and a bracket made of organic glass. Part B consists of circles of conducting copper wires (labeled as 12) and support made of organic glass as shown in Fig. 2. Besides, a function generator is used to produce a certain current IC for the Ampere force generator in the calibration process as shown in Fig. 1. The Ampere force can be expressed as FA = mBIC LC , determined by the number of turns m, magnetic induction B between parallel permanent magnet plates, and current IC and effective length LC of the conducting copper wires. The pulsed Ampere force produced by the pulsed Ampere force generator should be measured and checked before the application of the calibration device. Assembled with Part B, Part A of the generator can be weighed by an electronic mass balance (sensitivity: 10 μg, maximal mass: 80 g). When the current IC inside part B is adapted by the function generator, the digital reading of the weight of Part A changes because the supporting force from the electronic mass balance will decrease or increase owing to the direction of the Ampere force. Obviously, the variation of the digital reading represents the Ampere force produced by the generator under the current IC . The relationship between the current of the Ampere force generator and the Ampere force can be measured as shown in Fig. 3, and expressed as FA = AAmpere + BAmpere IC ,
(25)
Ampere force FA [μN]
1000
500
0
0
0.05 0.1 0.15 Current of Ampere force generator IC [A]
0.2
FIG. 3. Measured curves for Ampere force versus electric current.
where AAmpere = (0.007 ± 0.158) μN, and BAmpere = (5.73 ± 0.02) × 103 μN/A. Here, the linearity of the curve is sufficient for a calibration process. After obtaining the relationship of Eq. (25), Parts A and B are set onto the torsional arm and the X-Y two-dimensional displacement platform respectively as shown in Fig. 2. The distance between the axis of the torsional bar and the centerline of the Part A is denoted by L in Fig. 2(a) and is tested as (0.290 ± 0.001) m. Considering the difference between effective length R and L, the effective force impacting on the pendulum at the location of the thruster can be calculated as F (t) = FA (t)L/R.
(26)
When the pendulum is propelled by the Ampere force produced by the pulsed Ampere force generator, the pendulum will swing around the torsional bar. Especially when the Ampere force is produced repetitively with a same period as the vibration period of torsional pendulum the sympathetic vibration occurs. B. Measurement in single pulse mode
The impulse bit or single pulse produced by the ablative pulsed plasma thruster can be measured using the torsional pendulum. The calibration is conducted first. Given a certain current IC and duration τ A , the pulsed Ampere force generator produces the corresponding Ampere force and the known impulse bit Ib . As shown in Fig. 4, given a current IC of 197 mA and a duration of 0.1 s and considering the relationship given by Eq. (25), a pulsed Ampere force FA (t) with an amplitude of 1129.95 μN and a duration of 0.1 s can be produced. Together with Eq. (26), the corresponding impulse bit can be expressed as Ib = F(t)τ A = FA (t)τ A L/R = 0.5577 × 112.995 μN s = 63.0 μN s. Under this known impulse bit, the torsional pendulum can be propelled as shown in Fig. 4. In Fig. 5, the first peak PSD signal can be tested and labeled as Up, 0 . Under various impulse bits produced by pulsed Ampere force generator, the corresponding peak PSD signals can be obtained. The relationship between impulse bits and peak PSD signals is Up,0 = A0 + B0 Ib ,
(27)
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0.15
Pulsed Ampere force ends
Up,0 0
0
-0.05
0
2 Time [s]
4
-0.15
(c)
-0.15 0
where A0 = (−3.9 ± 253.0) × 10−5 V, B0 = (2.72 ± 0.34) × 10−4 V/μN s, the unit of Ib is μN s, and the unit of Up, 0 is V. The subscript 0 in Up, 0 indicates that the quantity Up, 0 is obtained in a calibration process. After calibration processes, the measurements of an impulse can be achieved. The time histories of PSD signals under different charged energies in the pulsed plasma thruster are listed in Fig. 6. The first peak PSD signals Up can be read out from the figure. Together with Eq. (27), the impulse bits under different charged energies (EC = 6.4, 12.1, 22.5 J) are obtained as shown in Fig. 7. Five measurements of impulse bits are conducted for every charged energy of capacitors in the experiments. The impulse bits increase with the charged energy of capacitors in the pulsed plasma thruster. The single pulse measurement mode of the thrust stand appears sufficient to measure thrust performance of the micro-thruster. However, when impulse bits decrease with charged energies, the corresponding torsional angles become extremely small and PSD signals become too small to be measured with high accuracy. In fact, when PSD signals decrease to orders of several mV or less, noise signals may partially cover the real PSD signals. As a result, the reading error of the PSD signals becomes more and more significant with a decrease in impulse bits. The detailed analysis of the noise signals in the thrust stands can be found in many past works.12 We do not intend to put emphasis on the analysis of the origin of complex noise signals in a thrust stand, though the noise of thrust stand is not negligible when PSD signals become small. Obviously,
Time [s]
5
10
FIG. 6. Time histories of PSD signals under different working conditions of the thruster. (a) EC = 6.4 J. (b) EC = 12.1 J. (c) EC = 22.5 J.
in single pulse measurement mode, measurement accuracy in the thrust stand will decrease with a decrease in impulse bits that we wish to measure.
C. Measurement in sympathetic resonance mode
According to the above analysis, it is difficult to test lowimpulse bits with high accuracy in single pulse mode. Therefore, higher precision and accuracy of measurement are expected according to the usage of sympathetic vibration in the thrust stand. To evaluate the ability to measure low-impulse bits produced by the thruster, the charged energy of the capacitors in the thruster is set at a low level of 6.4 J with the corresponding impulse bit is about 95 μN s as tested in single pulse mode as shown in Fig. 7. First, the calibration procedure is conducted by using the pulsed Ampere force generator as illustrated in Figs. 1 and 2. The function generator is used to control the Ampere force. Prior to setting the function generator, the period of the torsional pendulum is tested to be approximately 1.310 s. The period of the pulsed Ampere force is set to be the same as the period of the torsional pendulum for the formation of sympathetic vibration. Given the magnitude and duration of current in the pulsed Ampere generator, the certain impulse bits can be produced to propel the torsional arm. Fig. 8 shows two time histories of sympathetic vibrations produced by the pulsed 600
0.05
500
0.04
Impulse Bit Ib [μN⋅s]
Peak PSD signal Up,0 [V]
(b)
0
0
6
FIG. 4. A time history of PSD signal under a pulsed Ampere force.
0.03 0.02 0.01 0
0.15
0.15
-1000
Pulsed Ampere force starts -0.1 -2
-0.15 PSD signal [V]
PSD signal [V]
PSD signal
Ampere force [μN]
1000 τA=0.1 s
0.05
(a)
0
400 300 200 100
0
50 100 Impulse Bit Ib [μN⋅s]
FIG. 5. Peak PSD signals versus impulse bits.
150
0
0
5
10 15 Charged Energy EC [J]
20
25
FIG. 7. The impulse bit versus charged energy.
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0.3
1000
Pulsed Ampere Force
0
0.2
0 PSD Signal
-1000
-0.5
(b)
0.5
Pulsed Ampere Force
1000
0
0
-0.5 -5
PSD Signal 0
5
20
25
0 -0.1 -0.2
-1000 10 15 Time [s]
0.1
30
0
FIG. 8. Sympathetic vibrations produced by pulsed Ampere force generator. (a) Current amplitude: 100 mA, τ A = 0.1 s. (b) Current amplitude: 197 mA, τ A = 0.1 s.
Ampere force generator with different driving currents for the generator. The theoretical value of the magnification factor α(N) can be obtained from Eq. (23). It can be found that when β = 0, α(N) = N. However, the damping effect always exists, i.e., β > 0, therefore, α(N) < N. The damping effect with different β (β = 0, 0.01, 0.1, 0.5 Hz) and T = 1.310 s can be observed in Fig. 9. With an increase in β, the nonlinearity between the magnification factor and the peak number becomes more and more significant. From the calibration process as shown in Fig. 8, the magnification factor α(N) can be measured and is listed in Fig. 9. This indicates that though α(N) < N because of the damping effect, the nonlinearity is insignificant for the thrust stand within N < 20 as listed. When the torsional pendulum is working in a resonance mode, the peak voltage is tested to be 0.1 Hz, for example), the resonance amplification effect will be damped more intensively. Fortunately, the influence of nonlinearity can be overcome effectively by adding more calibration data points when the effect of nonlinearity becomes significant; i.e., though the
5 Time [s]
10
FIG. 10. Sympathetic vibrations produced by the thruster.
calibrated curve is nonlinear, the calibration data can be applied in the impulse measurement. The time histories of PSD signals can be tested for different impulses produced by the pulsed Ampere force generator, and the magnification factors can be demonstrated to be independent of the impulses. For every peak number N, the relationship between impulse bits and peak PSD signals can be obtained and expressed as follows: Up,0 (N ) = AN + BN Ib ,
(28)
where BN is regarded as a function of the peak number N, i.e., BN ≈ ηφα(N), and both BN and AN can be easily obtained from the calibration process. Equation (27) can be regarded as the special case of Eq. (28) when N = 1, and then sympathetic resonance mode degenerates into single pulse mode. Besides, it can be demonstrated that BN ≈ B0 α(N). By adapting the power supply and ignition controller, the period of impulse bits produced by the thruster can be changed to be same with the torsional period of the pendulum. The sympathetic vibration produced by the thruster are measured as shown in Fig. 10. Peak PSD signals Up (N) for different times of sympathetic vibrations can be read out. Note that Up, 0 (N) is the peak PSD signals in the calibration process without the thruster working and Up (N) is the peak PSD signals in follow-up measurement with the thruster working. The corresponding impulse bits can be obtained as shown in Fig. 11 by using the calibrated relationship between impulse bits and peak PSD signals given by Eq. (28). The variation of 150
0.3 Calibration Data Analytical Data
1 2
15
1- β=0 2- β=0.01 3- β=0.1 10 4- β=0.5 (Unit- Hz)
3
5 4 0
0
5
10 15 Peak number N
FIG. 9. Magnification factor versus peak number.
20
Peak PSD Voltage Up(N) [V]
α(N) or Up,0(N)/(ηφIb)
20
Peak PSD Voltage Impulse Bit 0.2
100
0.1
50
0
0
1
2
3
4 5 6 7 Peak Number N
8
9
Impulse Bit Ib [μN⋅s]
(a)
PSD signal [V]
PSD Signal [V]
0.5
Rev. Sci. Instrum. 84, 125113 (2013)
Ampere Force [μN]
125113-7
0 10
FIG. 11. Variation of peak PSD voltage and impulse bit with peak number.
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impulse bits with peak number N is obtained within ranges of 88–96 μN s as shown in Fig. 11. Besides, it can be found that an impulse bit of about 91 μN s is obtained when the number of peaks is raised to 9. As shown in Fig. 7, the impulse bits under the charged energy of 6.4 J are obtained with a range of 89.2–108.2 μN s. This indicates that the results in sympathetic resonance mode are in well accordance with the results in single pulse mode. The results obtained by the sympathetic resonance measurement method are reliable, indeed. Compared with single pulse mode, the advantages of sympathetic resonance method lie in following aspects: (1) Higher precision in measuring the low impulse bits is achieved for the sympathetic resonance method, which will be discussed in the section on Uncertainty analysis, and (2) the sympathetic resonance method makes it possible to measure extremely low-impulse bits, which are difficult to test for the single pulse method if the original thrust stand is not improved or if another high precision method is not applied. Besides, because of the simple operational principle and structure the sympathetic resonance method is much more convenient and inexpensive to be implemented than other high precision methods, such as the laser interferometer method. D. Uncertainty analysis
Uncertainties of measurements in single pulse and sympathetic resonance modes are analyzed for thruster with a charged energy of 6.4 J. The possible sources of uncertainty within impulse measurements are listed herein. 1. Uncertainty induced by fluctuations of impulse bits
In the above theoretical analysis, the impulses for every peak number are assumed to be the same. However, the interval and magnitude of the impulses or the fine structure of the thrust curve may vary for every real pulsed thrust. The irregular interval and magnitude of impulse bits lead to fluctuations of impulse bits when the thrust works in resonance mode. Thus, errors of measurements may be induced by fluctuations of the impulse bits. It is reasonable to assume that there is a fluctuation δIb (n) for impulse bit, i.e., Ib (n) = I b (n) + δIb (n), where I b (n) is the mean value of the impulse bit. It is assumed that I b (n) = a · n + δ(n), where a [μN s/pulse] is the speed for a downtrend or uptrend of impulse bits, and δ(n) is the random part of the fluctuations. The relationship between the peak angles and the impulse bits can be obtained as θp (N) = φ[α(N )I b + aN (N + 1)/2 + N n=1 δ(n)]. The measured value of the impulse bit can be obtained as I˜b (N) = θp (N)/(φα(N )). The relative errors of impulse bits induced by fluctuations of impulses bits can be expressed as Ib (N )/Ib (N ) = {aN [N + 1 − 2α(N)]/2 + N n=1 δ(n)}/[α(N )Ib (N )]. For an optimal pulsed plasma thruster, the long-term stability of thrust performance is expected and realizable without a significant tendency of increment or decrement of impulse bits, i.e., a ≈ 0 [μN s/pulse]. Besides, the random fluctuations of impulse bits (δ(n)) are expected to be sufficiently low in contrast with the impulse bit Ib , i.e., δ(n) Ib (n). There-
Rev. Sci. Instrum. 84, 125113 (2013)
fore, for an optimal thruster, the relative errors induced by the fluctuations of impulse bits are negligible, i.e., Ib (N)/Ib (N) ≈ 0. In this article, the variations of impulse bits are observed and it is reasonable to assume that |a| < 0.1 μN s/pulse. The errors induced by random fluctuations δ(n) can be estimated as Ib (N )/Ib (N ) = N n=1 δ(n)/α(N )Ib (N ). Considering the uniformly distributed random fluctuations, the summation of fluctuations δ(n) becomes zero when the peak number N extends to infinity. In addition, the errors induced by random fluctuation δ(n) of an impulse bit can be reduced with the increment of peak number and can be regarded as negligible when peak number N increases to 10. For a peak number N < 10 and an impulse bit of 100 μN s, the relative errors with the exception of the influence of random fluctuations δ(n) can be estimated as |Ib (N)/Ib (N)| ≈ |{aN[N + 1 − 2α(N)]}/[2α(N)Ib (N)]| < 0.5%. Obviously, errors induced by fluctuations of impulse bits are controllable for the sympathetic resonance method. By improving the stability of thrust performance, the errors induced by fluctuations of impulse bits become insignificant. 2. Uncertainty in displacement measurement of the He-Ne laser spot using a PSD sensor
Considering that the coefficient A0 in Eq. (27) can be negligible, Eq. (27) together with Eq. (24) can be changed to θ p = B0 Ib /η. The conversion ratio φ in Eq. (13) can be obtained as φ = B0 /η = 2.1 × 10−7 rad/μN s. The displacement resolution of the PSD sensor, i.e., δlPSD , is
