Journal Pre-proofs New Algorithms to estimate electron temperature and electron density with contaminated DC Langmuir probe onboard CubeSat Shyh-Biau Jiang, Tse-Liang Yeh, Jann-Yenq Liu, Chi-Kuang Chao, Loren C. Chang, Li-Wu Chen, Chung-Jen Chou, Yu-Jung Chi, Yu-Lin Chen, ChenKang Chiang PII: DOI: Reference:
S0273-1177(19)30833-6 https://doi.org/10.1016/j.asr.2019.11.025 JASR 14549
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
24 May 2019 4 November 2019 18 November 2019
Please cite this article as: Jiang, S-B., Yeh, T-L., Liu, J-Y., Chao, C-K., Chang, L.C., Chen, L-W., Chou, C-J., Chi, Y-J., Chen, Y-L., Chiang, C-K., New Algorithms to estimate electron temperature and electron density with contaminated DC Langmuir probe onboard CubeSat, Advances in Space Research (2019), doi: https://doi.org/ 10.1016/j.asr.2019.11.025
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New Algorithms to estimate electron temperature and electron density with contaminated DC Langmuir probe onboard CubeSat Shyh-Biau Jiang1,2, Tse-Liang Yeh1,2, Jann-Yenq Liu4,5, Chi-Kuang Chao4,5, Loren C. Chang4,5, Li-Wu Chen1,3, Chung-Jen Chou1, Yu-Jung Chi1, Yu-Lin Chen1, Chen-Kang Chiang1,3 1Institute
of Opto-Mechatronics Engineering, National Central University, Taoyuan City 32001, Taiwan of Mechanical Engineering, National Central University, Taoyuan City 32001, Taiwan 3ViewMove Technologies, Inc., Taoyuan City 32001, Taiwan 4Institute of Space Science, National Central University, Taoyuan City 32001, Taiwan 5Center for Astronautical Physics and Engineering, National Central University, Taoyuan City 32001, Taiwan 2Institute
Corresponding author Chen-Kang Chiang:
[email protected]
Abstract Three algorithms are developed to process the response current of pulsed Langmuir probes (PLPs) onboard satellites estimating ideal pulse current, residual voltage, and the electron temperature (Te) and electron density (Ne) in the ionosphere. The contaminated layer on the probe surface could result in LPs obtaining unreliable Te and Ne. The PLP is designed to solve this problem, and however, the rise time of pulse signals not approaching to zero will still bring some errors. Meanwhile, after each pulse scan, it requires a certain period for discharging, which affects the spatial resolution of PLP measurements. To resolve these problems, three algorithms estimate the ideal pulse response current, calculate the contaminated layer residual voltage to shorten the inter pulse scan period, and compute Te and Ne accordingly. Measurements of the DMSP (Defense Meteorological Satellite Program) are used to validate the performance of the proposed algorithms. Results show that the error of Te and Ne can be reduced down to 5% each, and the scanning rate can be increased by about 5-50 times. Keywords: Electron temperature; Electron density; Pulsed plasma probe; Contaminated layer; algorithm
1
1. Introduction A Langmuir probe (LP) is a typical instrument for satellite missions in measuring the electron temperature (Te) and electron density (Ne) in the ionosphere. Langmuir and MottSmith (1924) invented the LP to study gas discharges in vacuum plasma chambers. When a voltage is applied on LP in plasma, the current-voltage (I-V) curve can be used to derive Te and Ne (Langmuir and Mott-Smith, 1924; Mott-Smith and Langmuir, 1926).
LPs
onboard rockets and satellites have been employing to study ionospheric plasma structures and dynamics of Te and Ne (Reifman and Dow 1949; Boggess et al., 1959; Spencer et al., 1965; Brace et al., 1973; Oyama and Hirao, 1976; Brace, 1998; Oyam, et al. 2004; Cussac et al., 2006; Lebreton et al., 2006; Jiang et al., 2012; Lin et al., 2017; Chen et al., 2018). With advances in manufacturing technologies, the size of satellites has significantly decreased. Heidt et al. (2000) introduced a NanoSat standard called CubeSat that can be developed and launched at low cost, thereby facilitating ionospheric research in low earth orbits.
This results in LPs being widely used to probe the ionospheric Te and Ne.
However, since 1938, several scientists have found the contaminated layer problem through observations of the hysteresis of the I-V curve.
Two different distorted I-V curves
could be obtained by an LP, when the applied voltage is swept upward and then downward (cf. Van Berkel, 1938; Wehner and Medicus, 1952; Sturges, 1973; Winkler et al., 2000; Hirt et al., 2001).
Oyama (1976) used a mass spectrometer to detect contaminants on the
probe surface and noted that water was the main contaminant, and the equivalent circuit of the contamination layer is represented by a resistor and a capacitor connected in parallel. In addition, other substances such as N2 and O2 were detected too, and this makes it difficult to prevent the probe surface from a nitride- or oxide-contaminated layer in a high-energy plasma environment, and hard to predict accurate capacitance values of the contamination layer.
The hysteresis is caused by the charge, which is stored in the capacitor of
contaminated layer experiences the charge-discharge process.
To overcome the
interference of the contaminated layer in the LP measurement, Szuszczewicz and Holmes (1975) used a different voltage discrete pulse scan instead of a continuous triangular wave scan referred to as "pulsed Langmuir probe (PLP)."
Oyama et al. (2012) studied the time
interval of PLP pulses and proposed the pulse on/off time ratio of 1/99 case gives the same result as the uncontaminated probe characteristic curve. Oyama (2015a) reviewed the in situ measurement of Te in the ionosphere and plasmasphere utilizing DC LPs, and technical factors that should be taken into account for reliable measurement.
For example, the surface area of the satellite can no longer be 2
considered infinite, the probe surface may be contaminated, and the stray capacitance effect of the circuit should be taken in to account. Regarding the small satellite LP measurement problem, Johnson and Maler (1950) added a reference electrode with the same surface area and shape in addition to the probe electrode to avoid the Debye sheath (DS) on the satellite surface that affected LP measurement (Oyama et al. 2015b). Recently, Chiang et al. (2019) proposed the ground Debye sheath compensation (GDSC) algorithm to estimate the I-V voltage change of the satellite DS correctly and derive Te and Ne accordingly. In this paper, we present three algorithms to improve the Te Ne estimation and scanning rate to reduce the effect of the contaminated layer charging in LP measurement.
We first
develop a model to simulate the effect of the contaminated layer in various Te, Ne, and probe scan voltage.
In section 3, three algorithms are introduced and used to estimate
the ideal pulse response, residual voltage, and Te Ne. Finally, measurements of the Defense Meteorological Satellite Program (DMSP) are used to validate the performance of our algorithms.
2. Equivalent circuit and simulation
Figure 1. Equivalent circuit of pulse LP measurement Figure 1 shows the LP equivalent circuit containing the satellite surface, probe surface, and a contaminated layer above them.
When an LP is immersed in plasma and
measurement the I-V curve of the DS, for a huge satellite, owing to the large surface area ratio of the probe and satellite, the impedance of the satellite DS is extremely small and can be ignored.
However, the surface area of the cubic satellite is not extremely large, and the
DS on the satellite surface can no longer be ignored. 3
Due to the high conductivity of the
space plasma, the plasma is expressed by a conductor here. In order to focus on the problem of the contaminated layer, we simplified and ignored the influences of satellite body charging or photoemission current in the model.
The DS circuit is equivalent to a diode in
parallel with a capacitor (Shimoyama et al., 2012), where the capacitance is extremely small and is on the order of pico-farads per square centimeter (Chen, 2006).
The voltage-current
characteristic of the probe and the satellite DS satisfy the orbital motion limited (OML) theory (Mott-Smith et al., 1926; Rober, 2007). OML๐๐(๐ฃ๐๐ โ ๐ฃ๐๐) = ๐ผ๐๐ = ๐ผ๐๐๐ + ๐ผ๐๐๐
=
{
๐๐ด๐๐๐๐ ๐๐ด๐๐๐๐
๐พ ๐ต๐ ๐
(1 โ
2๐๐๐ ๐พ ๐ต๐ ๐
exp 2๐๐๐
(
๐ฝ๐๐
๐(๐ฃ๐๐ โ ๐ฃ๐๐)
)
๐พ ๐ต๐ ๐
๐พ ๐ต๐ ๐
โ ๐๐ด๐๐๐๐
โ๐(๐ฃ๐๐ โ ๐ฃ๐๐)
)
๐พ ๐ต๐ ๐
โ ๐๐ด๐๐๐๐
2๐๐๐ ๐พ ๐ต๐ ๐ 2๐๐๐
exp
(
1+
๐(๐ฃ๐๐ โ ๐ฃ๐๐)
(
๐พ ๐ต๐ ๐
) ) )
๐(๐ฃ๐๐ โ ๐ฃ๐๐ ๐พ ๐ต๐ ๐
, ๐ฃ๐๐ โค ๐ฃ๐๐
๐ฝ๐๐
(1)
, ๐ฃ๐๐ > ๐ฃ๐๐
OML๐ ๐๐ก(๐ฃ๐ ๐๐ก โ ๐ฃ๐๐) = ๐ผ๐ ๐๐ก = ๐ผ๐๐ ๐๐ก + ๐ผ๐๐ ๐๐ก
=
{
๐พ ๐ต๐ ๐
๐๐ด๐ ๐๐ก๐๐ ๐๐ด๐ ๐๐ก๐๐
(1 โ
2๐๐๐ ๐พ ๐ต๐ ๐
exp 2๐๐๐
(
๐ฝ๐ ๐๐ก
๐(๐ฃ๐ ๐๐ก โ ๐ฃ๐๐)
)
๐พ ๐ต๐ ๐
โ๐(๐ฃ๐ ๐๐ก โ ๐ฃ๐๐)
โ ๐๐ด๐ ๐๐ก๐๐
)
๐พ ๐ต๐ ๐
โ ๐๐ด๐ ๐๐ก๐๐
๐พ ๐ต๐ ๐ 2๐๐๐ ๐พ ๐ต๐ ๐ 2๐๐๐
(
exp
1+
๐(๐ฃ๐ ๐๐ก โ ๐ฃ๐๐)
(
)
๐พ ๐ต๐ ๐
๐(๐ฃ๐ ๐๐ก โ ๐ฃ๐๐) ๐พ ๐ต๐ ๐
, ๐ฃ๐ ๐๐ก โค ๐ฃ๐๐
)
๐ฝ๐ ๐๐ก
(2)
, ๐ฃ๐ ๐๐ก > ๐ฃ๐๐
Where e is the electron charge, kB denotes the Boltzmann constant, ฮฒ is the shape parameter of the contact surface with the plasma, ฮฒ=0 for the planar probe, ฮฒ=0.5 for the cylindrical probe, and ฮฒ=1 for the spherical probe. Apr is the electron collection region/area of probe, ne and ni (me and mi) are respectively the electron and ion density (mass), and Te is the electron temperature. stationary in the plasma.
The above equation is constructed assuming the probe is
For satellites orbiting in a planetary environment and moving at
velocities that exceed the sound speed, we need to modify the ion current equation Iipr and Iisat as Ii = eneAc(Vs+Vt) in Eqs. (1) and (2). (Oyama & Cheng 2013), where vs is the satellite velocity, vt is the thermal velocity of ions, and Ac is the effective ion collection area of probe or satellite.
The values calculated in the two ion current models presented above will not
differ much in magnitude and they are much smaller than the electron current, these corrections for the ion current that will not change the numeric of Ipr and Isat very much.
To
focus on the nature of the ness, we describe a simplified model, ignoring the surface charging created from photoelectric current.
Also, we assume that the uniform distributions
of satelliteโs conducting surface, which will bring some errors in estimating Te and Ne in real-time flight. Both sides of the contaminated layer collect many ions and electrons and form an 4
electric double-layer capacitor. This capacitor connects in parallel with the equivalent resistance of the contaminated layer. The capacitance value is proportional to the surface area and the dielectric constant of the contaminant, and is inversely proportional to the thickness of the contaminated layer. For a cylindrical LP with a diameter of 3 mm and a height of 20 cm, the contaminated layer capacitance and resistance are the order of several microfarads and several hundred kilo-ohms, respectively (Oyama, 1976; Oyama et al., 2012).
A DC voltage source represents the scanning voltage.
Theoretically, the
triangular wave of the scanning voltage (Vscan) forms with many shorts pulses, which changes suddenly in voltage.
However, the voltage change is limited by the driving current
and the stray capacitance, to raise the output voltage instantaneously is difficult. Here, we use the first-order rising voltage model, Vscan(1-exp(-t/ฯv)), to describe the rising of the scanning voltage. The effects of stray capacitance are expressed by a time constant ฯv.
When Vscan is changed from 0 to the target voltage, the response current will
overshoot then gradually reduces. The reduces curve is determined by the DS and the contaminated layer, which can be expressed by a time constant of the response current ฯc = CcRcRs(Rc+Rs) (Szuszczewicz and Holmes, 1975), where Rs is a simple Ohmic approximation to the sheath impedance and proportional to 1/Ne, Cc and Rc are the effective contamination capacitance and resistance.
ฯc is proportional to vscan and 1/Ne, when
Te=2500K and Ne = 106 cm-3, ฯc is the order of 1ms to 100ms.
Figure 2. Response current corresponding to 1-V step voltage at Te = 2000 K and Ne = 106 cm-3. The horizontal axis indicates the normalization time, that is, the magnification of 5
ฯc. The vertical axis indicates the (a) scan voltage and (b) response current. Figure (2a) shows the pulse voltage rising edge approximates the pulse voltage as an exponential function.
We use the normalized time constant, ฯc/ฯv, to quantify and study
how the stray capacitance and the contaminated capacitance affects the response current. The greater ฯc/ฯv, the shorter the rising time.
Figure (2b) shows the response current.
If
ฯc/ฯv is extremely large, the stray capacitance dynamics are much faster than the contamination dynamics, and the response current a is close to an ideal pulse response. When ฯc/ฯv is getting smaller, the response current starts deviating away from the ideal one, and the maximum value can no longer be regarded as the ideal pulse response current. The x-mark indicated on each curve is the maximum current, which is smaller than the ideal pulse response, under the influence of stray capacitance. still some error, due to the non-ideal rising time.
Even for a ฯc/ฯv of 1000, there is
Another problem with maximum current
sampling is that it is susceptible to the signal noise. To show the effect of noise on the maximum current mensuration, we add 60nA of noise to the current response when ฯc/ฯv = 100.
The data shows that noise causes the over-estimated response current when
obtaining the maximum current.
Besides, limited by the sampling frequency of the ADC, it
is difficult to measure the maximum value of the response current directly.
3. Algorithm Development and Verification A. Ideal pulse response current estimation As mentioned above, due to the effects of ฯc and ฯv, the ideal pulse response cannot be obtained from the sampling current data.
In this section, we derive the Ideal pulse
response current estimation (IPRE) algorithm to estimate the current value of the ideal pulse. First, the mathematical model of pulse response can be constructed on Eq. (3) I๐๐(๐ก) = (1 โ exp ( โ ๐ก ๐๐ฃ))(a + bexp (ct)) โ I๐๐(๐ก) = (๐ + ๐exp (๐๐ก)) ๐คโ๐๐ ๐๐ฃ โช ๐ก
(3)
where Irr (t) is the non-ideal pulse response current; Iri (t) is the ideal pulse response current; and a, b, and c = 1/ฯc are dynamic parameters of current decrease. Eq. (3) shows that Irr (t) approximates to Iri (t) when the scan voltage enters the steady state. By sampling Irr (t) after the output voltage settles down and substituting these values into the dynamic equation Iri (t) = a + b exp (ct). The ideal impulse response current Iri (0) = a + b can be calculated by system identification (ID). Assuming that the pulse voltage reaches the steady state after a 6
delay time Td>>ฯv, and three with a sampling time interval Tp response current samples will satisfy the following equation:
{ Let K =
๐ผ(๐๐) = ๐ + ๐(exp (๐๐๐)) ๐ผ(๐๐ + ๐๐) = ๐ + ๐(exp (๐๐๐)) ร exp (๐๐๐) ๐ผ(๐๐ + 2๐๐) = ๐ + ๐(exp (๐๐๐)) ร exp (2๐๐๐)
๐ผ(๐๐) โ ๐ผ(๐๐ + 2๐๐) ๐ผ(๐๐) โ ๐ผ(๐๐ + ๐๐)
=
1 โ exp (2๐๐๐)
1 โ exp (๐๐๐) .
(4)
Then,
(exp (c๐๐))2 โ ๐พ(exp (๐๐๐)) + (๐พ โ 1) = (exp (๐๐๐) โ (๐พ โ 1))(exp (๐๐๐) โ 1) = 0 for(exp (c๐๐) โ 1) โ 0,exp (๐๐๐) = (๐พ โ 1) ๐๐๐ ๐ =
ln (๐พ โ 1) ๐๐
Substituting c into Eq. (4) and converting into a matrix form gives exp (๐๐๐) ๐ผ(๐๐) 1 ๐ ๐ ( ) B โ ๐ผ ๐๐ + ๐๐ = 1 exp (๐(๐๐ + ๐๐)) ๐ โ ๐ด ๐ 1 exp (๐(๐๐ + 2๐๐)) ๐ผ(๐๐ + 2๐๐)
[
][
][ ]
[]
(5)
By applying the least-squares estimation (LSE) to solve the equation , the ideal impulse response current Iri (0) = a + b can be calculated. Still, the noise will interfere with the method mentioned above. To reduce noise interference, we divide 3 x m samples into m ensembles with sampling frequency fs = 1/ts = m (1/Tp). Each ensemble consists of three current samples separated by an interval Tp. The sum of the first, second, and third points of each group satisfies the following equation.
{
โ โ โ
๐โ1
=
๐โ1
๐=0 ๐โ1
๐=0
=
๐=0 ๐โ1
๐โ1
๐=0 ๐โ1
๐=0 ๐โ1
๐=0 ๐โ1
๐=0
๐=0
๐=0
โ๐
๐พโฒ
๐โ1
โ ๐ + โ ๐(exp (๐(๐๐ + ๐๐ก๐ ))) ๐ผ(๐๐ + ๐๐ + ๐๐ก๐ ) = โ ๐ + exp (๐๐๐)โ ๐(exp (๐(๐๐ + ๐๐ก๐ ))) ๐ผ(๐๐ + 2๐๐ + ๐๐ก๐ ) = โ ๐ + exp (2๐๐๐)โ ๐(exp (๐(๐๐ + ๐๐ก๐ ))) ๐ผ(๐๐ + ๐๐ก๐ )
๐
๐ผ(๐๐ + ๐๐ก๐ ) โ โ๐ = 0๐ผ(๐๐ + 2๐๐ + ๐๐ก๐ )
๐=0
โ๐ ๐ผ(๐๐ ๐=0
+ ๐๐ก๐ ) โ
โ๐ ๐ผ(๐๐ ๐=0
+ ๐๐ + ๐๐ก๐ )
=
1 โ exp (2๐๐๐) 1 โ exp (๐๐๐)
Eq. (6) can be converted into the following matrix form:
7
(6)
and ๐ = ln (๐พโฒ โ 1) ๐๐
1 exp (๐๐๐) ๐ผ(๐๐) 1 exp (๐(๐๐ + ๐๐)) ๐ผ(๐๐ + ๐๐) 1 exp (๐(๐๐ + 2๐๐)) ๐ผ(๐๐ + 2๐๐) ๐ ๐ โฒ โฎ โฎ โฎ ๐ต โ = โ ๐ดโฒ ๐ ๐ ๐ผ(๐๐ + (๐ โ 1)๐ก๐ ) 1 exp (๐(๐๐ + (๐ โ 1)๐ก๐ )) ๐ผ(๐๐ + ๐๐ + (๐ โ 1)๐ก๐ ) 1 exp (๐(๐๐ + ๐๐ + (๐ โ 1)๐ก๐ )) ๐ผ(๐๐ + 2๐๐ + (๐ โ 1)๐ก๐ ) 1 exp (๐(๐๐ + 2๐๐ + (๐ โ 1)๐ก๐ ))
[
][
]
[]
[]
(7)
The LSE method can be used to calculate the ideal pulse response current Iri (0) = a + b. As mentioned above, Irr approaches Iri after the pulse voltage stabilizes, we must wait for a delay time Td related to ฯv after Vscan is applied, and select an appropriate Tp relate to ฯc.
Here, we define the normalized start sampling delay, Tdn = Td/ฯv, and the normalized ID
period, Tpn = Tp/ฯc(1v).
We observe the trend of the simulated estimation error as Tdn and
Tpn change and choose their optimal values. Among them, ฯc (1v) is the 1-V impulse response
current
time
constant.
Figure 3. A trend plot that estimates the average error and standard deviation of the response current changes with different (a) Tpn and (b) Tdn, at a plasma condition of Te = 2000 K, Ne = 105 cm-3, ฯc/ฯv = 500, and current noise is ยฑ30 nA. Figure 3a shows the error changes with different Tpn and Tdn = 7, and Figure 3b shows the error changes with different Tdn and Tpn = 0.2. The lines indicate the response current estimation errors at pulse voltages of 1 V, 0.8 V, 0.4 V, -3 V, and -5 V. As shown in Figure 3a, the values of the average estimation error and standard deviation decrease as Tpn increases. When Tpn is less than 0.1, the estimation algorithm may diverge. When it is greater than 0.1, the algorithm converges, the average error is within 0.5%, the standard deviation is within 0.2%, and the variation is reduced. It is recommended to use Tpn = 0.2, 8
that is, two times the safety margin to ensure convergence. The current estimation error for this value can be within 0.5%. As shown in Figure 3b, the higher Tdn, the higher is the current estimation accuracy. When Tdn = 5, the error is suppressed to less than 0.5% and the standard deviation is within 0.3%. When Tdn > 7 the error will not decrease much. It is recommended to use Tdn = 5~7 to estimate the ideal response current.
Figure 4. LP I-V curve estimation using IPRE method and maximum estimation method. To verify the performance of the IPRE algorithm, Figure 4 shows the LP I-V curve estimation using the IPRE method and maximum estimation method. The blue line indicates the simulation data of Irr and Iri at Te = 2000 K, Ne = 105 cm3, ฯc/ฯv = 500. The green line and black line are the current response obtained by using the traditional maximum estimation method, with a noise of ยฑ30 nA and ยฑ100 nA. The upper and lower limits are the standard deviations of the maximum estimation method, corresponding to 8 different pulse scan voltages of -3 V to 0.5 V with a step of 0.5V. The red X point indicates the average of the response currents estimated using the IPRE algorithm with optimal values of Tdn = 5 and Tpn = 0.2 with a noise of ยฑ100 nA. The data show that a bias exists in the mean value estimated by the maximum value and that no bias exists in the IPRE algorithm.
B. Residual voltage estimation When Vscan is applied to the LP, the contaminated capacitor will charge up gradually through the response current, then discharge back through the DS resistor during the pulse interval. If the bleed time is insufficient, the contaminated layer will still have a residual 9
voltage and cause the hysteresis. scanning rate of LPs.
However, the longer the pulse interval is, the lower the
For the IPRE algorithm, the problem may be more severe since we
need 2Tpn second to sample the current dynamics.
To resolve the problem, we propose
the contaminated layer residual voltage estimation (CRVE) algorithm to shorten the inter pulsescan period.
First, a bisection approximation method is used to calculate the cross-
voltage of the contaminated capacitor by the response current.
Then we use the
exponential-fitting method to calculate the residual voltage at the start time of the next pulse.
Figure 5. The trend of response current drop of different pulses. Figure 5 shows the trend of the current response drop of different pulses; it is adjusted horizontally by the time axis so that the response currents of different pulses overlap when they fall to the same current in 5a. 5b plots the voltage difference of two contaminated layers across the probe and the satellite, charged and discharged for 2.5s.
The results show that
the response current drop curve follows the same curve regardless of the pulse voltage. The Vc-current curve is determined by Te, Ne, and surface area ratio of the probe and the satellite. Assuming that the Te and Ne are known, Eqs. (1) and (2) can be used to calculate the contaminated layer voltage with the discharge response current.
According to the
Kirchhoff's current law, the sum of the inflow current through the probe DS and the flow through the satellite DS should be equal to zero, i.e., Ipr = -Isat. ๐ผ๐ก๐๐ก๐๐(๐ฃ๐,๐๐,๐๐) = ๐ผ๐๐(๐ฃ๐๐ โ ๐ฃ๐๐,๐๐,๐๐) + ๐ผ๐ ๐๐ก(๐ฃ๐ ๐๐ก โ ๐ฃ๐๐,๐๐,๐๐) = 0 Let Vscan and the residual voltage of the contaminated layer (Vc) both zero.
(8) The objective
function is defined to be equal to the sum of the inflow currents of the two DSs,
๐ผ๐๐ ๐๐๐(๐ฃ๐๐) =
๐ผ๐ก๐๐ก๐๐(0,๐๐,๐๐) = ๐ผ๐๐( โ ๐ฃ๐๐,๐๐,๐๐) + ๐ผ๐ ๐๐ก( โ ๐ฃ๐๐,๐๐,๐๐). Here, we can found vpl by the bisection approximation method in appendix A.
Ipr and Isat can also be used to estimate vpr and vsat ๐ ๐๐ก Let ๐ผ๐๐ ๐๐๐(๐ฃ๐๐) = ๐ผ๐๐(๐ฃ๐๐ โ ๐ฃ๐๐,๐๐,๐๐) and ๐ผ๐๐๐
using the bisection approximation method. 10
(๐ฃ๐ ๐๐ก) = ๐ผ๐ ๐๐ก(๐ฃ๐ ๐๐ก โ ๐ฃ๐๐,๐๐,๐๐) be the objective function, and we make the objective function approximate Iobj to find vpr and vsat as given by the appendix B.
Finally, the residual voltage
of the contaminated layer can be calculated as vc = vpr โ vsat. During discharge, the capacitor voltage decreases exponentially, and the signal-tonoise-ratio SNR also decreases exponentially.
It is challenging to calculate vc directly when
the SNR is too small. Here, we use the curve when the voltage starts to discharge to identify the parameters of exponential decay, and then calculate vc(t).
By using the same
conditions as those in the ideal pulse response current estimation algorithm, sampling is started after the falling edge at Tdn =7, and Tpn =0.2. The current measurement sampling period, Ts, is reduced by m times compared to the system ID sampling period, i.e., Tp = mTs, to improve the anti-noise capability by using a multi-experimental method. Assume that the voltage dynamics satisfy the exponential function drop, V(t) = a exp(bt), where a and b are the parameters to be identified. Then, the sampled data satisfies the mathematical model: ๐(๐๐ + n๐๐ + ๐๐๐) = ๐(๐๐ + ๐๐๐ + (๐ โ 1)๐๐)exp (๐๐๐) + ๐๐๐๐๐(๐,๐)
(9)
where n is the experiment number and k is the identification cycle number. The equation is rewritten in matrix form as follows:
[
] [
][
๐(๐๐ + 0๐๐ + ๐๐) ๐(๐๐ + 0๐๐ ) ๐๐๐๐๐(0,1) ๐(๐๐ + 1๐๐ + ๐๐) ๐(๐๐ + 1๐๐ ) ๐๐๐๐๐(1,1) โฎ โฎ โฎ ๐(๐๐ + (๐ โ 1)๐๐ + ๐๐) = ๐(๐๐ + (๐ โ 1)๐๐ ) exp (b๐๐) + ๐๐๐๐๐(๐ โ 1,1) ๐๐๐๐๐(0,2) ๐(๐๐ + 0๐๐ + 2๐๐) ๐(๐๐ + 0๐๐ + ๐๐) โฎ โฎ โฎ
]
(10)
๐โ1 ๐
E(exp (b๐๐)) =
Let ๐ =
โ๐ = 0 โ๐ = 1๐(๐๐ + ๐๐๐ + ๐๐๐)๐(๐๐ + ๐๐๐ + (๐ โ 1)๐๐)
ln (๐ธ(exp (๐ ๐๐))) ๐๐
๐โ1 ๐ โ๐ = 0 โ๐ = 1๐2(๐๐
(11)
+ ๐๐๐ + (๐ โ 1)๐๐)
, and we find the value of exp(ฦ Ts) exp(ฦ Tp). Eq. (9) can be rewritten
as ๐(๐๐ + n๐๐ + k๐๐) = ๐(๐๐)(exp (๐๐๐ ))๐(exp (๐๐๐))๐ + ๐๐๐๐๐(๐,๐),
(12)
Its matrix form is as follows:
[
][
] [ ]
(exp (๐๐๐ ))1(exp (๐๐๐))0 ๐(๐๐ + 0๐๐ + 0๐๐) ๐๐๐๐๐(0,1) (exp (๐๐๐ ))1(exp (๐๐๐))0 ๐(๐๐ + 1๐๐ + 0๐๐) ๐๐๐๐๐(1,1) โฎ โฎ โฎ ๐(๐๐ + (๐ โ 1)๐๐ + 0๐๐) = (exp (๐๐๐ ))๐ โ 1(exp (๐๐๐))0 V(๐๐) + ๐๐๐๐๐(๐ โ 1,1) ๐๐๐๐๐(0,2) ๐(๐๐ + 0๐๐ + 1๐๐) (exp (๐๐๐ ))0(exp (๐๐๐))1 โฎ โฎ โฎ
11
(13)
๐โ1 ๐
๐(๐๐) = where
โ๐ = 0 โ๐ = 1๐(๐๐ + n๐๐ + k๐๐)((exp (๐๐๐ ))๐(exp (๐๐๐))๐)
(14)
2
๐โ1 ๐
โ๐ = 0 โ๐ = 1((exp (๐๐๐ ))๐(exp (๐๐๐))๐)
can be used to estimate the residual voltage of the pulse interval using
equation ๐(๐๐ + ๐๐๐ ) = ๐(๐๐)(exp (๐๐๐ ))๐.
The residual voltage estimation involves the
initial guess of Te and N, it is necessary to analyze the influence of Te and Ne of the vc estimation error to understand the applicable range of CRVE.
As long as the algorithm has
a sufficiently wide applicable range of variation in the estimated Te and Ne values, the I-V curve within this range can be improved by residual voltage estimation. The CRVE algorithm can then accurately estimate the residual voltage, Te, and Ne of the contaminated layer through iterations with a suitable initial guess of Te and Ne.
On LP measurement, we can
evaluate the effect of residual voltage by the ratio of residual voltage ๐ฃ๐ to Vscan.
The LP's
equivalent circuit implies that this voltage ratio is inversely proportional to the impedance of the Debye Sheath. IV curve is an exponential function from the ion saturation region to the transition region, which means that the Debye Sheath's impedance is smaller as the scan voltage increases. curves.
Figure5a shows that Vscan=1V has a higher voltage ratio than other
Due to inaccurate initial guesses of Te and Ne, we choose Vscan=1V to study the
estimated error of residual voltage in CRVE algorithm.
Figure 6. Contaminated layer residual voltage and CRVE correction result using incorrect initially estimated (a) Te and (b) Ne values. The X-axis is discharging time; we call t=0 when the pulse is ended. Figure 6a shows the estimation error with different initial guesses of Te. The dashed line represents ๐ฃ๐ after Vscan =1V is applied to the LP, then removed at t=0s. The simulation environment's Te = 2000 K and Ne = 105 cm-3.
The true Ne = 105 cm-3 and several different
Te (1500K~3500K) are substituted into the CRVE algorithm to obtain the estimated residual voltage ๐ฃ๐ during discharge.
The solid line represents the correction result ๐ฃ๐ โ ๐ฃ๐.
As
shown in the figure, when the initial guess is correct, the estimated error is zero. And 12
regardless of whether the initially estimated temperature is the lowest or highest ionospheric temperature, the estimation error is between ยฑ0.02 V, which is much lower than the residual voltage. It shows that the CRVE algorithm can effectively increase the accuracy of the I-V curve regardless of whether the initial guess Te is too large or small.
Figure 6b shows the
vc estimation error with different initial guesses of Ne. The dashed line represents the residual voltage after Vscan = 1 is applied to the LP for Te = 2000 K and Ne = 105 cm-3.
The
solid line represents the correction result, the correct Te = 2000 K and multiple different Ne from 0.7x105 cm-3(underestimation) to 10x105 cm-3 (overestimation) substituted into the CRVE algorithm.
When the initial Ne is overestimated, the correction result is between 0
and the residual voltage. When the guessed Ne is underestimated, the correction result is greater than 0. On the other hand, when the guessed Ne is less than 70% of original Ne, the estimate error (black line) is larger than the residual voltage (dashed line). Therefore, as long as the initial estimate is higher than the actual concentration or the underestimation of the initial estimate but not less than 70%, the CRVE algorithm improves the accuracy of the I-V curve.
C. Te and Ne estimation In this section, we derive a Te and Ne estimation algorithm that uses the minimum number of pulses, to further shorten the scanning rate of LP.
For each Te and Ne
measurement, multiple I-V response sample points must be measured to identify the parameters in the OML formula.
Figure 7. Four scan voltages of the I-V curve used to calculate Te and Ne. Figure 7 represents four pulse scan voltages used to calculate Te and Ne.
First, we
assume that the electron current in the ion saturation region of the I-V curve is approaching 13
to zero that can be regarded as the ion current, and use Vscan1 and Vscan2 to identify mathematical model parameters of the ion current.
Then, the ion current is subtracted from
the current to obtain the electron current at Vscan3 and Vscan4 in the transition region. Te and Ne can be obtained by the electron current equation. (2), the ion current is determined by ฮฒ.
Finally,
According to Eqs. (1) and
For ฮฒ = 0 or 1, the ion curve is a straight line.
Here,
we use a linear mathematical model Ii(vpr) = a vpr + b to describe the ion current of the I-V curve.
A straight line requires at least two points to identify the parameters a and b.
Therefore, there must be at least two negative pulse response samples in the ion saturation region.
[
] [
][ ] [ ] [ ]
๐ผ๐(๐ฃ๐ ๐๐๐1) ๐ฃ๐ ๐๐๐1 1 ๐ ๐ ๐ I = ๐ผ (๐ฃ = ๐ฃ = ๐ ๐ , ๐ = ๐ โ1๐ผ ๐ ๐ ๐๐๐2) ๐ ๐๐๐2 1 ๐
(15)
Because the current measurement is inevitably subject to noise interference and deviates from the true value, if the difference between the two negative voltages selected is larger, the effect of the response current error on the slope of the line is smaller. A more accurate ion current can be estimated by extrapolating this line. Moreover, the farther the electron current in the ion saturation region is from the transition region, the closer it is to zero; therefore, the selection circuit can provide the maximum negative voltage, vscan1=-5v, as one of the scanning voltages. The closer vscan2 is to the transition region, the farther it is from the first negative voltage and the better is the extrapolation effect. However, we have previously assumed that the electron current at vscan2 is almost zero, but the closer it is to the transition region, the greater the electron current.
14
Figure 8. Estimates of average and standard deviation error of Ii(0) for extrapolation with vscan1 = -5 V, vscan2 = -4.9~0 V, Te = 2000 K, Ne = 105 cm-3, noise= ยฑ30 nA. We investigate the effect of vscan2 on the extrapolation, corresponding to the Ii(0), to provide a reference for selecting the negative voltage of the second pulse.
In simulation
experiments, we have the correct Ii(vscan) data. By substituting different vscan2 from -4.9V to 0V into Eq 12, we can obtain multiple estimated value of Ii(0), then it subtract with the correct Ii(0) to get multiple Ii(0) error. Ii(0) error.
Figure 8 plots the average and the standard deviation of the
The average Ii(0) error does not change with vscan2 from -4.9 V to -1 V and then
gradually increases as it enters the transition region due to the Ii(vscan2) is no longer close to zero.
The standard deviation decreases as the difference between vscan2 and vscan1
increases. The upper envelope of the standard deviation has a minimum at vscan2 =-1.6V, which is the optimum second pulse voltage. If the interval whose value is smaller than the minimum value by +20% is defined as the minimum value interval, the minimum interval will be between -1.1 V and -2.3 V. The minimum interval for different Te and Ne values are presented in appendix C, and the range of values is seen to partially overlap. Here, we choose vscan1=-5v and vscan2=-2v as the optimal scan voltage. The electron current I(vpr-vpl) in the transition region increases exponentially as vpr-vpl increases,
where
๐ผ(๐ฃ๐๐ โ ๐ฃ๐๐) = ๐(exp (๐(๐ฃ๐๐ โ ๐ฃ๐๐))), ๐ = ๐๐ด๐๐๐๐ 15
๐พ ๐ต๐ ๐
2๐๐๐,
๐
c = ๐พ ๐ต๐ ๐ .
The
electron current can be obtained by subtracting the ion current from the measurement current in the transition region. and the transition region ion current can be estimated by linear extrapolation of the current from the ion saturation region. The electron flow exponential function model has two parameters b and c; therefore, at least two sampling points, vscan3 and vscan4, are needed to identify the parameters. Assume that the floating potential vscan3 = 0 is selected as the third sampling point to save the time waiting contaminated layer discharge. Let the fourth scan voltage be vscan4 = vd. The response electronic current equations of the third and fourth scan voltages satisfy the following:
{ ๐ผ( โ ๐ฃ๐๐ + ๐ฃ๐) ๐ผ( โ ๐ฃ๐๐)
๐ผ( โ ๐ฃ๐๐) = ๐ exp (๐( โ ๐ฃ๐๐)) ๐ผ( โ ๐ฃ๐๐ + ๐ฃ๐) = ๐exp (๐(๐ฃ๐ โ ๐ฃ๐๐)) = ๐ผ( โ ๐ฃ๐๐)exp (๐๐ฃ๐)
= exp (c๐ฃ๐)โ ๐ = ln
(
๐ผ( โ ๐ฃ๐๐ + ๐ฃ๐) โ ๐ฃ๐๐
)/๐ฃ
๐
. The parameter ๐ =
๐ผ( โ๐ฃ๐๐) exp (๐)
(16) can be solved by
substituting the obtained parameter ๐ into the response electron current equation of the ๐
third scan voltage. ๐ and ๐ can be substituted into Eq. (1) to solve ๐๐ = ๐พ๐ต๐ and ๐
๐๐ = ๐๐ด๐๐
2๐๐๐๐ ๐
.
When the satellite impedance is not considered, the higher vscan4 is selected; the larger the response current is and the further away from vscan3 = 0 it is, the better the identification result is. However, if the impedance of the satellite DS is considered, the potential of the satellite DS will be in the ion saturation region when the transition region of the probe DS is scanned, and its impedance is extremely large. Therefore, as vscan4 increases, most of the scan voltage will be shared by the satellite DS and the occupancy ratio of the probe DS voltage will decrease.
The probe DS voltage is the primary signal we need to estimate Te
and Ne. Satellite voltage is an interference that distorts the I-V curve.
In this study, we
select vscan4 as two times the probe DS voltage, vscan4 ~ 0.4 v, which ensured that the probe voltage was higher than the satellite voltage.
Appendix D shows the scan voltage that was
two times the probe DS crossover voltage, where Te = 1500 K, 2000 K, and 2500 K and Ne = 105 and 106 cm-3.
We show a complete Te and Ne iterative estimation algorithm in
Figure9 and Appendix E, which is consist of the IPRE, CRVE, and GDSC algorithms to solve the small area ratio and the contamination problem.
16
Figure 9. The flowchart of the iterative algorithm.
17
Figure 10 DMSP data, Te, and Ne estimated using the Te and Ne iterative estimation algorithm. The pulse on time is 80 ms, and the on/off time ratio is 1/0.5. Ts=100 us. The Te and Ne measured by DMSP are treated as true values to test the performance of the Iterative Algorithm. The surface area of the LP onboard the simulating CubeSat is 58.2 cm2, with satellite/probe area ratios of 6. response current.
The noises with ยฑ30 nA are added to the
The black curves in Figures 10a and 10b indicate the measured values
of Te and Ne during 11:32-12:01 7 December 2017. The estimation errors of Te and Ne are shown in Figures 10c and 10d.
The estimates based on the uncorrected I-V curves are
shown in blue, which used the method introduced in Section C to estimate Te and Ne only, without IPRE, CRVE, and GDSC.
For the uncorrected curves, Te is overestimated by
about 20-50%, and Ne underestimated by 10-20%.
The green curves show the result of
Te and Ne estimated method from Section C and GDSC but without IPRE, CRVE, which can reduce the offset error. The results by implementing all the algorithms discussed above in red line can further reduce the error down to 5% each, all the pulse time and pulse interval time used in three estimation lines are the same.
4. Discussion and conclusion Three algorithms are designed to resolve the contamination problem of DC LPs. The 18
contaminations on the surface of LPs and satellites cause the hysteresis distortion and result in incorrect Te and Ne measurements.
We build an equivalent circuit model to reproduce
the contamination problem in the time constant ฯc.
The model shows that when the surface
area of satellites and probes decrease or Ne increase, the ฯc becomes smaller and causes more severe distortion, which matches the "density dependences" description (Oyama 1976).
Moreover, when the ฯc is getting short and close to ฯv, it is difficult to obtain the
accurate ideal pulse response of DS.
The IPRE algorithm was designed to solve this
problem by identifying the current drop curve.
Figure 4 shows that the IPRE algorithm can
calculate the response current more accurately and improve noise immunity than the maximum method.
Traditionally, we need to set a long discharge time between pulses to
avoid the effects of residual voltage.
According to the literature of Oyama et al. (2012), it
takes about 1/99 of the on/off ratio to allow the residual voltage to discharge completely. For the IPRE algorithm, the discharge time may become longer due to the sampling of the current dynamics.
To overcome this problem, we derive the CRVE algorithm, which by
estimating and compensating the residual voltage, so that the LP can perform next scan without waiting fully discharge. The algorithm takes a Tp (i.e., 0.2 ฯc) time to collect the dynamic discharge curve of the contaminated layer to ensure the accuracy of the estimation of the residual voltage. Simulation experiments with DMSP data show that Te and Ne can be calculated effectively even if the on/off time ratio is only 1/0.5.
Meanwhile, a calculation
method of Te and Ne is purposed, which requires only three pulse scanning voltages to be applied.
When ฯc =100 ~ 10 ms, the scanning rate of Te and Ne measurement is about
1/ (3*3Tpn ฯc) = 5 ~ 50 Hz.
In the case of obtaining the distortion curve but unknowing Te
and Ne onboard small satellite. We put forward an iteration Algorithm which consists of IPRE, CRVE, and GDSC algorithm to solve the contaminant and the small area ratio problem. Finally, we used data from the DMSP as a truth Te and Ne to verify the effectiveness of the algorithm. The simulation data show that the proposed algorithm can reduce the error of Te and Ne down to 5% each.
Acknowledgements This study has been supported by Ministry of Science and Technology (MOST), MOST 107-2119-M-008-023.
We thank Dr. FY Chang in Center
for Astronautical Physics and Engineering (CAPE), National Central University for providing the data information of DMSP. The authors thank reviewers for their useful comments and suggestions, which lead to a better presentation of this paper.
19
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Appendix A. ๐๐ 1. Choose ๐ฃ๐๐(0,0), ๐ฃ๐๐(0,1), S.T. ๐ผ๐๐ ๐๐๐(๐ฃ๐๐(0,1)) ร ๐ผ๐๐๐(๐ฃ๐๐(0,0)) < 0, ๐ฃ๐๐(0,2) = 0.5(๐ฃ๐๐(0,0) + ๐ฃ๐๐(0,1)), ๐ = 0 2. ๐ฃ๐๐(๐ + 1,1) = ๐ฃ๐๐(๐,2) ๐๐ ๐ผ๐น: ๐ผ๐๐ ๐๐๐(๐ฃ๐๐(๐,1)) ร ๐ผ๐๐๐(๐ฃ๐๐(๐,2)) < 0, ๐ฃ๐๐(๐ + 1,0) = ๐ฃ๐๐(๐,1) ๐ธ๐ฟ๐๐ธ: ๐ฃ๐๐(๐ + 1,0) = ๐ฃ๐๐(๐,0) 3. ๐ผ๐น:|๐ผ๐๐ ๐๐๐(๐ฃ๐๐(๐,2))| < ๐, result ๐๐๐ = ๐ฃ๐๐(๐,2), go to end. 4. i = i + 1 go to 2.
Appendix B. ๐๐๐ 1. Choose ๐ฃ๐๐(0,0), ๐ฃ๐๐(0,1), S.T. ๐ผ๐๐๐ ๐๐๐ (๐ฃ๐๐(0,1)) ร ๐ผ๐๐๐ (๐ฃ๐๐(0,0)) < 0 ๐ฃ๐๐(0,2) = 0.5(๐ฃ๐๐(0,0) + ๐ฃ๐๐(0,1)), ๐ = 0 2. ๐ฃ๐๐(๐ + 1,1) = ๐ฃ๐๐(๐,2) ๐๐๐ ๐ผ๐น:(๐ผ๐๐๐ ๐๐๐ (๐ฃ๐๐(๐,1)) โ ๐ผ๐) ร (๐ผ๐๐๐ (๐ฃ๐๐(๐,2)) โ ๐ผ๐) < 0, ๐ฃ๐๐(๐ + 1,0) = ๐ฃ๐๐(๐,1) ๐ธ๐ฟ๐๐ธ: ๐ฃ๐๐(๐ + 1,0) = ๐ฃ๐๐(๐,0) 3. ๐ผ๐น:|๐ผ๐๐๐ ๐๐๐ (๐ฃ๐๐(๐,2)) โ ๐ผ๐| < ๐, result ๐๐๐ = ๐ฃ๐๐(๐,2), go to end 4. i = i + 1 go to 2.
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Appendix C. Te(K) 1500 -3 5 Ne(cm ) 10 106 Minimum -0.8~-2 -1.1~-2.2 interval (V) Appendix D. Te(K) Ne(cm-3) Vscan4(V)
2000 106
105
-1.1~-2.3
-1.5~-2.6
-1.4~-2.6
1500 105 0.46
2500
105
2000 106 0.46
105 0.62
2500 106 0.62
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106 -1.8~-2.9
105 0.77
106 0.77
Appendix E. Te and Ne Iterative Algorithm 1. For ๐ฃ๐ ๐๐๐(๐,๐ก), ๐ = 1โฆ4 ๏ฌ Sample the simulation responses currents ๐ผ๐๐(๐ฃ๐ ๐๐๐(๐,๐ก),๐),๐คโ๐๐๐ ๐ = 1โฆ4, ๐ = 1โฆ4 ๏ฌ Estimates the ideal pulse response ๐ผ๐๐(๐ฃ๐ ๐๐๐(๐,๐ก)) with sampled response ๐ผ๐๐(๐ฃ๐ ๐๐๐(๐,๐ก),1),๐ผ๐๐(๐ฃ๐ ๐๐๐(๐,๐ก),2) 2. Initial guess of ๐๐(0,๐ก) = ๐๐(๐ก โ 1), and ๐๐(0,๐ก) = ๐๐(๐ก โ 1). k=0 3. For ๐ฃ๐ ๐๐๐(๐,๐ก), ๐ = 1โฆ4 ๏ฌ Estimates the residual cross voltage ๐ฃ๐๐๐ (๐ฃ๐ ๐๐๐(๐,๐ก)) with sampled response ๐ผ๐๐ (๐ฃ๐ ๐๐๐(๐,๐ก),3),๐ผ๐๐(๐ฃ๐ ๐๐๐(๐,๐ก),4) and ๐๐(๐,๐ก) ๐๐(๐,๐ก) ๏ฌ Correct the pulse voltage ๐ฃ๐๐๐(๐,๐ก) = ๐ฃ๐ ๐๐๐(๐,๐ก) โ ๐ฃ๐๐๐ (๐ฃ๐ ๐๐๐(๐ โ 1,๐ก)), where ๐ฃ๐๐๐ (๐ฃ๐ ๐๐๐(0,๐ก)) = ๐ฃ๐๐๐ (๐ฃ๐ ๐๐๐(4,๐ก โ 1)) ๏ฌ Refer to the algorithm in (2) to calculate the ๐ฃ๐๐(๐ฃ๐ ๐๐๐(๐,๐ก)) and ๐ฃ๐ ๐๐ก(๐ฃ๐ ๐๐๐(๐,๐ก)), with given ๐ฃ๐๐๐(๐,๐ก), ๐๐(๐,๐ก) and ๐๐(๐,๐ก) 4. Extrapolation to calculates the ion current ๐ผ๐๐๐(๐ฃ๐ ๐๐๐(3,๐ก)), ๐ผ๐๐๐(๐ฃ๐ ๐๐๐(4,๐ก)) with given ๐ผ๐๐๐ (๐ฃ๐ ๐๐๐(1,๐ก)) = ๐ผ๐๐(๐ฃ๐ ๐๐๐(1,๐ก)), ๐ผ๐๐๐(๐ฃ๐ ๐๐๐(2,๐ก)) = ๐ผ๐๐(๐ฃ๐ ๐๐๐(2,๐ก)) 5. Calculate the electron current ๐ผ๐(๐ฃ๐ ๐๐๐(3,๐ก)) = ๐ผ๐๐(๐ฃ๐ ๐๐๐(3,๐ก)) โ ๐ผ๐๐๐(๐ฃ๐ ๐๐๐(3,๐ก)), ๐ผ๐(๐ฃ๐ ๐๐๐(4,๐ก)) = ๐ผ๐๐(๐ฃ๐ ๐๐๐(4,๐ก)) โ ๐ผ๐๐๐(๐ฃ๐ ๐๐๐(4,๐ก)) ๐๐ต๐๐
2๐๐๐,
๐
6.
Identify for ๐ = ๐๐ด๐๐๐๐
๐ = ๐๐ต๐๐ with given ๐ผ๐(๐ฃ๐ ๐๐๐(3,๐ก)),
7. 8.
๐ผ๐(๐ฃ๐ ๐๐๐(4,๐ก)), ๐ฃ๐๐(๐ฃ๐ ๐๐๐(3,๐ก)) , ๐ฃ๐๐(๐ฃ๐ ๐๐๐(4,๐ก)). Calculates the ๐๐(๐ + 1,๐ก), and ๐๐(๐ + 1,๐ก) with known ๐ and ๐. If ( |๐๐(๐ + 1,๐ก) โ ๐๐(๐,๐ก)| > ๐๐ or |๐๐(๐ + 1,๐ก) โ ๐๐(๐,๐ก)| > ๐๐) then k=k+1 and go to 3.
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