High temperature calibration and testing method based on phase method displacement sensor

Abstract

The application discloses a high-temperature calibration and testing method based on a phase method displacement sensor and belongs to the technical field of measurement and testing. The application takes each working frequency point in and out of the transmission frequency band of the sensor, completes the calibration of each working frequency point at different temperatures, compensates the measurement deviation caused by the transmission cable in the high-temperature environment by using the out-of-band measurement value, compensates the measurement deviation caused by the thermal expansion of the sensor cable and the probe material, the change of the dielectric property of the transmission medium and the spatial filtering effect in the high-temperature environment by increasing the stopband frequency point and introducing a high-temperature compensation factor function, and simultaneously proposes a repeated iteration correction algorithm for the high-temperature compensation factor and the calibration formula, so that the high-precision displacement testing of the sensor in the high-temperature environment is realized. The application can realize the accurate testing of the phase method displacement sensor under the condition that the temperature of the sensor probe is unknown, improve the measurement capability of the sensor in the high-temperature environment, and guarantee higher stability, measurement precision and reliability.

Description

Technical Field

[0001] This invention relates to a high-temperature calibration and testing method based on a phase-based displacement sensor, belonging to the field of metrology and testing technology. Background Technology

[0002] Displacement sensors, as a common and fundamental type of sensor, offer high measurement accuracy and have a wide range of applications. Currently, the main directions for further development of displacement sensors are improving measurement accuracy, reducing size, and adapting to more specialized working conditions and environments. For example, the size of the blade tip clearance inside an aero-engine significantly affects the engine's efficiency and stability. However, due to the special working environment of the engine, including high temperatures, exhaust gases, and fuel contamination, a specialized displacement sensor and effective calibration and testing methods are needed to achieve high-precision measurement of the blade tip clearance under these conditions.

[0003] The main principle of a displacement sensor based on the phase method is as follows: a continuous wave signal emitted by a signal source is radiated towards the target through the sensor. The reflected signal is reflected back to the sensor at the metal end face of the target. By measuring the phase difference between the reflected signal and the measured signal, the displacement information of the target is obtained. When the sensor operates at frequency f, the formula for calculating the displacement value R corresponding to the phase difference φ is:

[0004]

[0005] This method is a non-contact measurement, which reduces the impact on the operating state of the measured object and has good environmental adaptability. To avoid phase ambiguity, the measured displacement must be within half the wavelength range of the sensor's operating frequency.

[0006] When the operating temperature of a displacement sensor increases, the thermal expansion of materials causes all internal components to enlarge, lengthening the continuous wave transmission path and leading to measurement deviations. Secondly, the transmission cable and probe contain a transmission medium; the dielectric constant of this medium increases at high temperatures, shortening the signal wavelength and severely impacting the sensor's final phase measurement results. For example, in aero-engine blade tip clearance testing, the gas temperature in the high-pressure compressor can reach 600°C, while in the high-pressure turbine it can reach 1100°C or even higher, posing a challenge to achieving high-precision testing at high temperatures. Furthermore, the installation conditions within the engine require the sensor probe to be as small as possible, making it difficult to integrate a temperature sensor for real-time monitoring of the operating environment temperature. Therefore, achieving effective calibration and high-precision testing of displacement sensors in all high-temperature environments without using a temperature sensor remains a bottleneck in the development of displacement sensors operating in high-temperature environments.

[0007] US6856281B2 discloses a calibration method for a phase-based testing system, proposing a dual-frequency testing method using the sensor's passband and stopband to compensate for phase deviations caused by cable elongation at high temperatures. While this invention offers many valuable insights, its drawbacks include: overly idealized theoretical assumptions that neglect the elongation of the sensor probe itself at high temperatures, and the phase deviation caused by changes in the dielectric properties of the transmission medium at high temperatures. Summary of the Invention

[0008] To address the issues of large measurement errors and poor accuracy caused by high-temperature testing environments in phase-based displacement sensors, this invention aims to provide a high-temperature calibration and testing method for phase-based displacement sensors. This method compensates for measurement deviations caused by thermal expansion of sensor cable and probe materials, changes in the dielectric properties of the transmission medium, and spatial filtering effects under high-temperature conditions by adding a stopband frequency and introducing a high-temperature compensation factor function. Simultaneously, it proposes a repetitive iterative correction algorithm for the high-temperature compensation factor and calibration formula, achieving high-precision displacement testing of the sensor under high-temperature environments. This invention enables real-time displacement measurement in high-temperature environments without requiring temperature measurements, ensuring higher stability, measurement accuracy, and reliability.

[0009] The objective of this invention is achieved through the following technical solution:

[0010] This invention discloses a high-temperature calibration and testing method based on a phase-based displacement sensor, comprising the following steps:

[0011] Step A: Measure the transmission frequency band range of the sensor. Take one operating frequency point within the sensor's transmission frequency band (passband) and one outside the frequency band (stopband). During the sensor calibration process, simultaneously measure the phase difference between the transmitted and reflected signals at the passband and stopband frequencies. The purpose of adding the stopband frequency measurement is to compensate for the measurement phase shift caused by the lengthening of the transmission cable and changes in the dielectric properties of the medium under high temperature conditions.

[0012] Step B: Using all the calibration data from the sensor calibration process in Step A, i.e., the phase difference values ​​at all calibration points and under temperature conditions, calculate the high-temperature compensation factor for each calibration point, forming a high-temperature compensation factor function related to the standard displacement value, which is used to correct the compensation factor during the sensor test data processing.

[0013] Step C: Establish a high-temperature calibration formula for the sensor. The high-temperature compensation factor is introduced into the calibration formula to compensate for the measurement phase shift caused by the thermal expansion of the probe material, the change in the dielectric properties of the transmission medium, and the spatial filtering effect at high temperatures.

[0014] The testing method further includes step D: substituting the sensor's room temperature calibration data into the calibration formula in step C to obtain the corresponding measured displacement value. Then, based on steps B and C, the sensor's high-temperature compensation factor and calibration formula are revised, and the iteration is repeated, with the measured displacement value being revised repeatedly, until the measured displacement values ​​of two adjacent measurements meet the set threshold relationship. At this point, the sensor test data is determined to be stable, achieving high-precision displacement testing of the sensor.

[0015] Further, step A includes: Step A1, measuring the transmission frequency band range of the sensor, finding the frequency point with the highest transmission efficiency within the transmission frequency band range as the passband operating frequency point of the sensor, assuming that the sensor can transmit the entire signal outward through the probe at this frequency point; selecting a frequency point outside the transmission frequency band range as the stopband operating frequency point of the sensor, assuming that the sensor signal cannot pass through the probe at the stopband operating frequency point and can only be completely reflected back to the transmission cable at the root of the probe; Step A2: on the sensor calibration platform, the signal source simultaneously emits continuous wave signals at the passband and stopband operating frequencies to the sensor probe, and simultaneously records the passband phase difference φ measured by the sensor at the two frequency points at all calibration points within the measurement range. c and stopband phase difference φ b Complete the room temperature calibration of the microwave sensor; Step A3: Under the high temperature environment of the sensor, simultaneously record the sensor passband phase difference φ at all calibration points within the measurement range. 1c and stopband phase difference φ 1b The high-temperature calibration of the microwave sensor was completed.

[0016] Further, step B includes: step B1, obtaining the passband phase difference value φ at room temperature according to step A2. c and stopband phase difference φ b The passband phase difference φ obtained in step A3 at high temperature 1c and stopband phase difference φ 1b And the formula k = (φ 1c -φ c ) / (φ 1b -φ b ) Calculate the high temperature compensation factor k for the corresponding calibration point. When there are two or more temperature points, k represents the slope of the curve formed by the relationship between the sensor passband measurement phase change and the stopband phase change caused by temperature. Step B2: Organize the high temperature compensation factor k values ​​on all calibration points and fit them to form the sensor high temperature compensation factor function: k = g(r), where r represents all standard displacement values ​​in the calibration process.

[0017] Furthermore, in step C, the calibration formula between the direct measurement result of the sensor and the standard displacement is established as: r = f(φ) c -kφ b ), where φc φ b The passband and stopband phase differences are measured by r under standard displacement, respectively, and k is a high-temperature compensation factor. The introduction of k compensates for the phase deviation of the sensor probe caused by thermal expansion and dielectric constant changes, as well as spatial filtering effects at high temperatures.

[0018] Further, step D shown includes: step D1, setting the initial value of the high-temperature compensation factor k, that is, the average value of k at all calibration points. Substituting the calibration formula from step C, we get: The formula should include the sensor's room temperature calibration data from step A2, i.e., the measured values ​​φ at all calibration points. c and φ b Where r is the standard displacement value of the corresponding calibration point, the corresponding calibration curve f1 is fitted; Step D2: During the sensor test, the measured values ​​of the sensor passband and stopband are... and Substitute the calibration formula f1 from step D1 to calculate the corresponding displacement value. Step D3: Substitute the r1 calculated in Step D2 into the high-temperature compensation factor function in Step B2 to correct the high-temperature compensation factor k, resulting in: k1 = g(r1); Step D4: Substitute the high-temperature compensation factor k1 calculated in Step D3 into the sensor calibration formula in Step C to correct, forming: r = f2(φ c -k1φ b Step D5: Fit the corrected sensor calibration curve; Step D2: Combine the measured values ​​of the sensor passband and stopband obtained in step D2. and Substitute the calibration formula f2 from step D4 into the recalculated sensor displacement value: Step D6: Compare whether the difference between r1 and r2 meets the set threshold requirement. If it does, the measurement result is determined to be stable and reliable, and r2 is the final displacement value measured. If it does not meet the threshold requirement, the measurement result is determined to be unstable. Replace r1 with r2 and enter it into step D3. Repeat steps D3-D5.

[0019] Beneficial effects:

[0020] This invention provides a high-temperature calibration and testing method for phase-based displacement sensors, solving the problems of large measurement errors and poor accuracy caused by high-temperature testing environments. This method enables accurate testing of phase-based displacement sensors under conditions where the sensor probe temperature is unknown, improving the sensor's measurement capabilities in high-temperature environments.

[0021] 1. This invention discloses a high-temperature calibration and testing method based on a phase-based displacement sensor. This method, by adding real-time measurement of the stopband frequency signal and combining it with the sensor's high-temperature compensation factor function, achieves real-time measurement and compensation of the displacement effect caused by changes in the sensor's temperature environment at any measurement time. Since this method does not require measuring the real-time temperature value during sensor testing, it enables accurate testing of phase-based displacement sensors under conditions where the sensor probe temperature is unknown.

[0022] 2. This invention discloses a high-temperature calibration and testing method based on a phase-based displacement sensor. It establishes a calibration model that combines the sensor's passband and stopband signals with a high-temperature compensation factor function. Furthermore, it provides an iterative correction algorithm for data processing based on the sensor's direct measurement data. Because this method effectively compensates for measurement deviations caused by thermal expansion of sensor cable and probe materials, changes in the dielectric properties of the transmission medium, and spatial filtering effects under high-temperature conditions, it can improve the measurement accuracy and capability of the phase-based displacement sensor under high-temperature environments. It can be used for the calibration and testing of phase-based displacement sensors operating in high-temperature environments. Attached Figure Description

[0023] Figure 1 This is a flowchart illustrating the method described in an embodiment of the present invention;

[0024] Figure 2 This is a schematic diagram of the working state of the microwave sensor for measuring engine blade tip clearance in the method described in this embodiment of the invention;

[0025] Figure 3 This is an example of a high-temperature compensation factor correction curve for a microwave sensor in the method described in this embodiment of the invention.

[0026] Figure 4 This is an example of a preliminary calibration curve for a microwave sensor in the method described in this embodiment of the invention;

[0027] Labeling explanation: 1-Transmission cable, 2-Sensor probe, 3-Engine casing, 4-Blade, 5-Signal source and test circuit. Detailed Implementation

[0028] The preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings, which form part of this application and, together with the embodiments of the present invention, serve to illustrate the principles of the present invention.

[0029] Figure 1 This is a flowchart illustrating the method described in this invention example, which is for the calibration and testing of displacement sensors that can operate in high-temperature environments.

[0030] Figure 2This is a schematic diagram illustrating the working state of the microwave sensor for testing engine blade tip clearance in the method described in this invention. The overall structure of the microwave sensor includes a transmission cable 1 and a probe 2. When the sensor is working, the probe 2 is inserted into the engine casing 3, and the end face of the probe 1 is retracted into the casing 3, facing the rotating blade 4. The engine blade tip clearance represents the radial distance between the inner wall of the casing 3 and the tip of the blade 4. Accurate measurement and reasonable design of the blade tip clearance are beneficial to reducing engine leakage and improving working efficiency. Within the sensor's transmission frequency band, the signal source 5 emits a continuous microwave signal, which is radiated towards the tip of the blade 4 through the cable 1 and the sensor probe 2, and reflected back to the sensor probe 2 at the blade tip surface. The signal then returns to the processing circuit 5 through the cable 1. The phase difference between the reflected signal and the emitted signal changes with the change in blade tip clearance, and the engine blade tip clearance is tested based on this principle.

[0031] Combined with appendix Figure 2 To the attached Figure 4 The method described in the embodiments of the present invention will be described in detail below. The sensor calibration method is as follows:

[0032] Step A: Select one operating frequency point within the sensor's transmission frequency band (passband) and one outside the frequency band (stopband). Simultaneously calibrate the sensor at both the passband and stopband operating frequency points. The addition of the stopband operating frequency point is used to compensate for the measurement phase deviation caused by the transmission cable under high temperature conditions.

[0033] Step B: Use sensor calibration data to generate a high-temperature compensation factor function. The high-temperature compensation factor function is used to compensate for the measurement phase deviation caused by thermal expansion of probe material, change of dielectric properties of medium and spatial filtering effect at high temperature.

[0034] Step C: Combine the high temperature compensation factor function to establish a high temperature calibration model for the sensor.

[0035] Sensor measurement methods include:

[0036] Step D: Set an appropriate threshold, and based on the high-temperature calibration model established in Steps B and C, repeatedly correct and iterate the high-temperature compensation factor and calibration formula on the sensor test data until a stable test result is obtained, thus obtaining a high-precision displacement measurement value of the sensor under high-temperature conditions.

[0037] In step A, a suitable passband operating frequency f is selected based on the actual transmission frequency band of the sensor being measured. c and stopband operating frequency f b Because at the passband frequency f cThe signal is radiated from the sensor probe to the blade tip and reflected back to the sensor at the blade tip surface. Therefore, the signal path is: cable length L from the signal source to the root of the sensor probe, sensor probe length D, and blade tip clearance distance R. Thus, the sensor passband test phase value φ... c It can be represented as:

[0038]

[0039] Similarly, due to the operating frequency f at the sensor stopband... b The signal is reflected at the root of the sensor probe and only travels the length L of the cable from the signal source to the sensor. Therefore, there is a sensor passband test phase value φ. b for:

[0040]

[0041] In the above two equations, λ b , λ c These are the passband and stopband signal transmission wavelengths in vacuum, respectively, ε rL ε rD Let be the relative permittivity values ​​of the medium in the cable and the sensor probe, respectively, where it is assumed that the medium in the sensor probe is uniformly distributed.

[0042] When the ambient temperature rises, the main changes in the sensor are the increase in structural dimensions of the cable and probe due to thermal expansion and the increase in the dielectric constant of the transmission medium, which are manifested as the cable and probe lengths increasing to L1 and D1, respectively, and ε rL and ε rD Increase to ε respectively rL1 and ε rD1 Therefore, the passband and stopband phases obtained by the test circuit at high temperature are respectively:

[0043]

[0044]

[0045] In step B, the high-temperature compensation factor function is defined and calculated first. Using formulas (2) and (4), we have:

[0046]

[0047] Using formulas (3) and (5), we have:

[0048]

[0049] make:

[0050]

[0051] Then we have:

[0052] φ 1c -kφ 1b =φ c -kφ b (9)

[0053] Let k be defined as the high-temperature compensation factor for the sensor, where k represents the slope of the curve formed by the change in the sensor's passband phase and the change in its stopband phase caused by temperature rise. Simplifying equation (8), we get:

[0054]

[0055] Since the elongation at high temperatures is relatively small compared to the actual length of the sensor probe, we set D1 / D≈1. Since the length of the sensor cable is much greater than the probe length, we similarly have L1 / L≈1. Therefore, equation (10) can be simplified to:

[0056]

[0057] Where λ b / λ c And D / L is a constant. The ratio of the increase in dielectric constant of the two materials at high temperature can be approximated as a constant. However, in actual testing, the compensation factor k is not a constant. Due to the spatial filtering effect, the high-temperature compensation factor varies under different blade tip clearances. By compiling the high-temperature compensation factor k values ​​under all standard clearances, the high-temperature compensation factor function k = g(r) of the sensor is obtained by fitting.

[0058] On a certain calibration platform, a sensor sample was calibrated at room temperature and high temperature. The obtained measurement data is shown in Table 1. The sensor's passband operating frequency is 24.214 GHz, the stopband operating frequency is 23.6 GHz, and the calibration range is 0.1-5 mm. Six calibration points were taken within this range.

[0059] Table 1. Room temperature and high temperature calibration data for microwave sensor examples.

[0060]

[0061] The ambient and high temperature measurement data of each calibration point were collected, and the high temperature compensation factor k under different standard gaps was calculated according to formula (8). The curves of k versus standard gap were obtained and attached. Figure 3 As shown, the high-temperature compensation factor function for the sensor is formed by fitting:

[0062] k = g(r) = 0.008 r 5 - 0.081 r 4 + 0.22 r 3 - 0.18 r2 + 0.623 r + 0.969(12)

[0063] In step C, a high-temperature calibration model for the sensor is established. Analysis using formula (9) shows that, under a defined standard blade tip clearance, the calculated f = φ from the sensor's passband and stopband measurement data is... 1c -kφ 1b The value is constant and unaffected by the sensor probe temperature. Therefore, the formula for the high-temperature calibration model of the sensor is determined as: r = f(φ c -kφ b In the formula, r is the standard clearance value, and φ is... c φ b These are the phase values ​​measured in the passband and stopband of the sensor at the calibration point, respectively.

[0064] The following section describes in detail the high-temperature testing process of microwave sensors, using a sensor calibration model.

[0065] In step D, the average value of k at all calibration points is taken as the initial value. In this example... Substituting this into the sensor calibration model, the initial calibration formula is obtained as follows:

[0066] r=f1(φ c -1.79φ b (13)

[0067] In the formula φ c and φ b This includes all room temperature measurement data in Table 1. The calibration curve generated after fitting is shown in the appendix. Figure 4 As shown in the figure, the sensor measurement values ​​are converted from phase values ​​to corresponding gap values. It can be seen that the calibration curve in this example is a fifth-order polynomial curve.

[0068] High temperature measurement value of sensor and Substitute the result into the calibration formula f1, then substitute the result into formula (12) to correct the high-temperature compensation factor k value, and then further correct the calibration formula using the k value, repeating this iteratively. Continue until the results of two adjacent cycles are close and reach the threshold requirement, then the measurement result is considered stable and reliable, ultimately achieving accurate testing of the engine blade tip clearance at high temperatures. Table 2 shows the final blade tip clearance measurement results and measurement errors obtained in this example. It can be seen that the maximum error is 47 μm. The main measures to reduce the measurement error are to increase sensor calibration points and improve the correction function of the high-temperature compensation factor k.

[0069] Table 2 Measurement error of microwave sensor at high temperature (unit: mm)

[0070] 0.1 -1.1792 0.1000 0.0000 1 -0.3878 1.0003 -0.0003 1.5 0.0137 1.4982 0.0017 2 0.4736 2.0004 -0.0004 2.5 1.0647 2.5420 -0.042 3 1.3407 2.9998 0.0002 3.5 1.7349 3.4966 0.0034 4 2.0748 3.9988 0.0012 5 2.8260 4.9983 0.0017

[0071] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A high-temperature calibration and testing method based on a phase-based displacement sensor, characterized in that: Includes the following steps, Step A: The area within the sensor's transmission frequency band is called the passband; the area outside the sensor's transmission frequency band is called the stopband. Measure the range of the sensor's transmission frequency band by taking one operating frequency point within the sensor's transmission frequency band and one outside the sensor's transmission frequency band. During the sensor calibration process, simultaneously measure the phase difference between the transmitted and reflected signals at the passband and stopband frequencies. Step A specifically includes the following steps. Step A1: Measure the range of the sensor's transmission frequency band, select the frequency point with the highest transmission efficiency within the range as the sensor's passband operating frequency point; select a frequency point outside the range as the sensor's stopband operating frequency point. Step A2: On the sensor calibration platform, the signal source simultaneously emits continuous wave signals at both the passband and stopband operating frequencies to the sensor probe. Simultaneously, the passband phase difference measured by the sensor at both frequencies is recorded at all calibration points within the measurement range. and stopband phase difference Complete the room temperature calibration of the sensor; Step A3: Under the high temperature environment of the sensor, simultaneously record the sensor passband phase difference values ​​at all calibration points within the measurement range. and stopband phase difference Complete the high-temperature calibration of the sensor; Step B: Using all the calibration data from the sensor calibration process in Step A, i.e., the phase difference values ​​at all calibration points and under temperature conditions, calculate the high-temperature compensation factor for each calibration point to form a high-temperature compensation factor function related to the standard displacement value. Step C: Establish a high-temperature calibration formula for the sensor. A high-temperature compensation factor is introduced into the high-temperature calibration formula to compensate for the measurement phase shift caused by the thermal expansion of the probe material, the change in the dielectric constant of the transmission medium, and the spatial filtering effect at high temperatures. Step D: Substitute the sensor's room temperature calibration data into the sensor's high temperature calibration formula in step C to obtain the corresponding measured displacement value; Then, based on steps B and C, the high-temperature compensation factor and high-temperature calibration formula of the sensor are revised again, and the iteration is repeated, and the measured displacement value is revised again until the measured displacement value between two adjacent measurements meets the set threshold relationship. Then, the sensor test data is determined to be stable, and high-precision displacement testing of the sensor is achieved.

2. The high-temperature calibration and testing method based on a phase-based displacement sensor as described in claim 1, characterized in that: Step B includes the following steps: Step B1: Obtain the passband phase difference value at room temperature according to step A2. and stopband phase difference The passband phase difference value obtained in step A3 at high temperature and the stopband phase difference at high temperature and formula Calculate the high-temperature compensation factor k for the corresponding calibration point; Step B2: Organize the high-temperature compensation factor k values ​​at all calibration points and fit them to form the sensor's high-temperature compensation factor function: In the formula, r represents all standard displacement values ​​during the calibration process.

3. The high-temperature calibration and testing method based on a phase-based displacement sensor as described in claim 2, characterized in that: The specific implementation method of step C is as follows: The high-temperature calibration formula for the sensor, which establishes the relationship between the sensor's direct measurement results and the standard displacement, is as follows: In the formula , These are the passband phase difference and stopband phase difference values ​​obtained by measuring r under standard displacement values, respectively, and k is the high-temperature compensation factor. The introduction of k compensates for the measurement phase shift caused by thermal expansion and dielectric constant changes, as well as spatial filtering effects, in the sensor probe at high temperatures.

4. The high-temperature calibration and testing method based on a phase-based displacement sensor as described in claim 3, characterized in that: Step D includes the following steps: Step D1: Set the initial value of the high-temperature compensation factor k, which is the average value of the high-temperature compensation factor k over all calibration points. Substituting the sensor high-temperature calibration formula from step C, we get: Substitute the sensor's room temperature calibration data from step A2, i.e., the passband phase difference values ​​at all calibration points. and stopband phase difference Let r be the standard displacement value of the corresponding calibration point, and fit the corresponding calibration curve. ; Step D2: During sensor testing, measure the passband phase difference and stopband phase difference of the sensor. and Substitute the values ​​into the calibration formula from step D1 to calculate the corresponding displacement values. ; Step D3: Calculate the result in step D2. Substituting the high-temperature compensation factor function from step B2, we correct the high-temperature compensation factor k to obtain: ; Step D4: Apply the high-temperature compensation factor calculated in step D3. Substituting the sensor high-temperature calibration formula from step C into the correct formula, we obtain: The corrected sensor calibration curve is then fitted. Step D5: Measure the sensor passband phase difference and sensor stopband phase difference values ​​obtained in step D2. and Substitute the calibration formula from step D4 Recalculate the sensor displacement value: ; Step D6: Comparison and If the difference meets the set threshold requirement, the measurement result is considered stable and reliable. That is, the final measured displacement value; if it does not conform, the measurement result is determined to be unstable, and... Alternative Substitute into step D3 and repeat steps D3-D5.

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