CN105157914B - A kind of system and method in internal combustion engine cylinder pressure signal time domain gyration domain - Google Patents

Abstract

The invention discloses a system and method for time-domain conversion of an internal-combustion engine cylinder pressure signal to an angle domain, a cylinder pressure measurement device and a pulse signal measurement device. The cylinder pressure measurement device and pulse signal measurement device collect the in-cylinder pressure signal and The pulse signal corresponding to the flywheel ring gear is transmitted to the internal clock acquisition system, and the internal clock acquisition system further transmits the received data to the processor for processing; the precise zero-crossing point of the pulse signal is determined by an interpolation algorithm, and on this basis, the in-cylinder The pressure signal is converted from the time domain to the angle domain; this method effectively avoids the problem of false triggering due to the loss or interference of the encoder signal in the traditional test method, and effectively improves the reliability of the data acquisition process.

Description

A system and method for time-domain conversion of internal-combustion engine cylinder pressure signal to angle domain

technical field

The invention relates to a test method for a pressure signal in a cylinder of an internal combustion engine, in particular to a system and a method for converting the pressure signal in a cylinder of an internal combustion engine from a time domain to an angle domain based on an interpolation algorithm.

Background technique

The internal combustion engine cylinder pressure signal contains a wealth of information about the combustion process in the cylinder, which can provide important references for the development and performance improvement of internal combustion engines. Every 720-degree crankshaft angle of the internal combustion engine is a working cycle, and the collection and analysis of the pressure signal in the cylinder is based on the crankshaft angle. For this reason, the existing in-cylinder pressure signal testing systems mostly use external clock sampling to test the in-cylinder pressure. This test method requires the use of an encoder to provide a trigger signal, which is used to control the test system to collect data at a specific angle step. During use, the housing of the encoder needs to be connected to the body of the engine, and the rotary shaft of the encoder is connected to the crankshaft of the engine. After the encoder is fixed, when the engine is working, the crankshaft drives the encoder shaft to rotate, and the encoder converts the rotation of the crankshaft into a pulse signal through the principle of photoelectric conversion, which is used to trigger the acquisition system to collect the pressure signal in the cylinder.

This method needs to find a suitable position to fix the encoder at the free end of the engine. Due to the large number of components and high integration at the free end of the engine, it is difficult to fix the encoder. The encoder shaft and the crankshaft rotate synchronously, requiring high concentricity between the two. If there is a deviation between the centers of the two, it is easy to cause the loss of the encoder output signal, or even the breakage of the encoder shaft, which will adversely affect the reliable operation of the entire test system. . In addition, the output of the encoder is a high-frequency pulse signal, which is sent to the test system for shaping and processing to trigger sampling. There are various interferences such as dynamometers in the engine test site, which may easily lead to false sampling in the test system.

According to the above-mentioned analysis of the existing in-cylinder pressure signal test methods, it can be seen that the existing methods have problems such as inconvenient installation of the sensor, and are easily affected by interference; and because the encoder is difficult to install, the current method can only be applied to laboratory testing occasions , for vehicle engines, due to the limited space at the free end of the engine, it is difficult to test the in-cylinder pressure.

Contents of the invention

In order to solve the deficiencies in the prior art, the invention discloses a method for testing the pressure signal in a cylinder by using an internal clock acquisition system, and then converting the collected time-domain signal into an angle-domain signal through an interpolation algorithm. This method no longer uses the encoder to trigger sampling, but installs the pulse signal test sensor at the end of the engine flywheel, and simultaneously collects the pressure signal and pulse signal in the cylinder through the internal clock acquisition system. After the acquisition is completed, interpolation processing is performed on the pressure signal in the cylinder based on the pulse signal to realize the conversion of the time domain signal and the angle domain signal. The pulse signal sensor used in this method only needs to be installed near the ring gear of the flywheel, which has low installation requirements and is easy to implement; the internal clock sampling method is used to collect signals, which eliminates the problem of abnormal sampling caused by the external clock sampling method being easily triggered by interference signals. .

To achieve the above object, the specific scheme of the present invention is as follows:

A system for turning the pressure signal in the cylinder of an internal combustion engine in the time domain to the angle domain, comprising: a cylinder pressure measuring device and a pulse signal measuring device, and the cylinder pressure measuring device and the pulse signal measuring device correspond the collected in-cylinder pressure signal to the flywheel ring gear The pulse signal is transmitted to the internal clock acquisition system, and the internal clock acquisition system further transmits the received data to the processor for processing;

In the processor, the point at which the output voltage of the pulse signal measuring device is zero is obtained through the interpolation algorithm, that is, the zero-crossing point. According to the data of the zero-crossing point, the time interval between two adjacent teeth is obtained, and the correspondence between any angle between the two teeth is obtained through the interpolation algorithm. The data.

Further, the cylinder pressure measuring device includes a cylinder pressure sensor, which is installed on the cylinder head of the engine and connected to the combustion chamber of the engine through a pressure measuring channel.

Further, the pulse signal measuring device is installed at the flywheel end of the engine, and the front end of the pulse signal measuring device is about 1 mm away from the tooth top of the flywheel ring gear. When the engine is working, when each tooth of the flywheel ring gear passes the sensor, the sensor will be triggered. Generate a pulse signal.

Furthermore, the pulse signal measuring device adopts a magnetoelectric sensor.

Further, the internal clock acquisition system is a multi-channel high-speed data acquisition card working in an internal clock sampling mode, and the data transmission mode of the multi-channel high-speed data acquisition card adopts a USB interface or a PCI interface high-speed data transmission interface to communicate with the processor.

A method for turning an internal-combustion engine cylinder pressure signal into an angle domain in a time domain, comprising:

The point at which the output voltage of the pulse signal measuring device is zero is obtained by an interpolation algorithm, that is, the time Tstart and Tend of the zero crossing point; the angle mark signal is the signal tested by the pulse signal measuring device;

The duration between the two moments Tstart and Tend is the duration corresponding to one tooth of the engine flywheel ring gear, and the corresponding angle between the two moments Tstart and Tend is the duration angle θ of one tooth;

According to the two moments of Tstart and Tend, the continuous angle θ of one tooth and the angle interval θ _{interpolation interval} after converting the pressure signal in the cylinder from the time domain to the angle domain, determine the time step T between adjacent points of the cylinder pressure signal in the angle domain _step ;

According to the sampling interval T _step , the current tooth start time Tstart and the angle deviation T _off between the first angle domain point and the current tooth start time Tstart obtained by interpolation of the current tooth, it is converted into the corresponding time T _n in the angle domain, n =0,1,2,3...;

According to the time T _n corresponding to the obtained angle domain, n=0, 1, 2, 3..., it is transformed into the pressure value y _n in the cylinder of each point in the angle domain, n=0, 1, 2, 3... before and after passing through this point The measured time-domain in-cylinder pressure signal is obtained by linear interpolation.

Further, when the point at which the output voltage of the pulse signal measuring device is zero, that is, the zero-crossing point, is obtained through an interpolation algorithm, the three-time Newton interpolation method is used to first obtain two points before and after the zero-crossing point, (x ₀ , y ₀ ), (x ₁ , y ₁ ), (x ₂ , y ₂ ) and (x ₃ , y ₃ ), among them, x ₀ ~ x ₃ are the sampling numbers of each sampling point, since the four points are continuous sampling points, each adjacent The intervals between the points differ by 1 sampling point time; while y ₀ ~ y ₃ refers to the pulse signal sampling results of two points before and after the zero crossing point. All x and y in the Newton interpolation formula are reversed and used, and y is equal to zero to obtain the corresponding the x-value.

Further, the corresponding angle θ between the time Tstart and Tend: θ=360°CA/number of teeth of the flywheel ring gear.

Further, the interval time difference is T _step :

In the formula, θ is the continuous angle of one tooth of the flywheel ring gear; the _{interpolation interval} of θ is the set angle interval between two adjacent points transformed into the angle domain; T _start and T _end are the zero-crossing time data obtained by interpolation.

Further, the expression of the time T _n (n=0, 1, 2, 3...) corresponding to each time in the angle domain is:

T _n = T start + n × T _step + T _off

In the formula, _Toff is the angle deviation between the first angle domain point obtained by the interpolation value of the current tooth and the starting time Tstart of the current tooth.

Beneficial effects of the present invention:

1) The present invention proposes a new method for installing a pulse signal sensor near the ring gear of the engine flywheel to realize the acquisition of the pressure signal in the cylinder. The pulse signal sensor has low installation requirements and is easy to implement; it effectively solves the difficulty in fixing the encoder when the encoder is used in the traditional cylinder pressure test method and the difficulty in ensuring that the encoder shaft and the crankshaft are highly concentric.

2) The cylinder pressure signal testing method proposed by the present invention first uses the internal clock acquisition method to collect the cylinder pressure signal and pulse signal as an analog signal, and then determines the precise zero-crossing point of the pulse signal through an interpolation algorithm, and realizes the cylinder pressure signal on this basis. The internal pressure signal is converted from the time domain to the angle domain; this method effectively avoids the problem of false triggering due to the loss or interference of the encoder signal in the traditional test method, and effectively improves the reliability of the data acquisition process.

Description of drawings

Fig. 1 is the hardware structure schematic diagram of the present invention;

Figure 2(a) is the comparison curve of the pressure signal and pulse signal in the cylinder measured for 5 cycles;

Fig. 2 (b) is the signal graph that time domain in-cylinder pressure signal is transformed into angle domain by the interpolation algorithm that the present invention proposes;

Fig. 3 is the zero-crossing moment schematic diagram of the pulse signal sensor that sampling process of the present invention obtains;

Fig. 4 is the time-domain rotation angle-domain principle of the in-cylinder pressure signal of the present invention.

detailed description:

The present invention is described in detail below in conjunction with accompanying drawing:

The present invention proposes a new method for testing the in-cylinder pressure signal in view of the problems of inconvenient installation of the encoder, loss of the encoder signal or being affected by interference and easy sampling abnormality in the current in-cylinder pressure signal testing method. The method has the advantages of convenient sensor installation, reliable operation, etc., can meet the needs of internal combustion engine research and development and production unit cylinder pressure signal test and analysis, and has broad application prospects.

The invention proposes a test method and system suitable for collecting pressure signals in cylinders, including two parts: a hardware system and a test analysis method.

1. Hardware system introduction: The hardware system includes two parts: the sensor and the internal clock data acquisition system. The sensor includes the pressure sensor used to test the pressure signal in the cylinder and the pulse signal test sensor installed near the flywheel ring gear. The cylinder pressure sensor is installed on the cylinder head of the engine, and is connected to the combustion chamber of the engine through a pressure measurement channel, and the output of the sensor is an analog signal. The pulse signal sensor is installed facing the teeth of the flywheel ring gear. When the engine is running, when each tooth of the flywheel ring gear passes the sensor, it will trigger the sensor to generate a pulse signal, which converts the cylinder pressure signal from the time domain to the angle domain. time base.

The basis of the method proposed by the present invention is to find the exact moment of the zero-crossing point through an interpolation algorithm, which requires several sampling points before and after the zero-crossing point. At present, the commonly used pulse signal sensors mainly include: magnetoelectric type, Hall type and photoelectric type. The signals output by the latter two types of sensors only have two levels, high and low. , the magnetoelectric sensor is the best choice for this method.

The internal clock acquisition system is a multi-channel high-speed data acquisition card that works in the internal clock sampling mode. This type of acquisition card uses its own clock signal to trigger the analog-to-digital conversion unit to achieve data acquisition. Because this method requires the acquisition card to have a higher sampling speed, the data transmission mode of the acquisition card needs to adopt high-speed data transmission interfaces such as USB interface or PCI interface. The signals of the cylinder pressure sensor and the pulse signal sensor used in this method are all sent to the high-speed data acquisition card, and each signal is regarded as an analog quantity for data acquisition, and the collected data is processed by the following method.

2. Introduction to the test analysis method: In order to convert the data sampled by the internal clock into data based on the crank angle, the following two steps are required: 1) Obtain the zero-crossing point of the pulse signal sensor through an interpolation algorithm; 2) Take the zero-crossing point as the benchmark , using the interpolation algorithm to convert the time-domain in-cylinder pressure signal to obtain angle-based data. The contents of each part are described below:

1) Acquisition of zero crossing point

The zero-crossing point is the point at which the output voltage of the pulse signal sensor is 0. The time between two adjacent zero-crossing points of the pulse signal is the duration of a tooth. This time plays an important role in the time domain pressure signal rotation angle domain. . Since the acquisition process is collected once at a certain interval, the data of each zero-crossing point of the pulse signal sensor cannot be completely obtained only through the data acquisition process. Here, the values of the sampling points before and after the zero crossing are used to obtain the exact moment of the zero crossing through an interpolation algorithm. Figure 3 is a schematic diagram of the pulse signal sensor obtained during the sampling process. The solid square in the figure represents the sampled data, and the circle represents the position of the zero crossing point.

The present invention uses interpolation to determine the position of the zero-crossing point, adopts multiple interpolation algorithms such as Newton interpolation, Lagrange interpolation and Hermit interpolation, and compares each method with different orders, and finds that the results obtained by different methods are similar, here The method of determining the zero-crossing point is explained by the third-order Newton interpolation method. The three-time Newton interpolation method needs to use two points before and after the zero-crossing point for calculation. The four points are as follows: (x ₀ , y ₀ ), (x ₁ , y ₁ ), (x ₂ , y ₂ ) and (x ₃ , y ₃ ), where x ₀ ~ x ₃ are the sampling numbers of each sampling point, since the four points are continuous sampling points, the intervals between adjacent points differ by 1 sampling point moment; and y ₀ ~ y ₃ It refers to the sampling results of two points before and after the zero crossing point, therefore, x ₀ ~ x ₃ and y ₀ ~ y ₃ are all known quantities.

The Newton interpolation formula is expressed as follows:

y＝N _n (x)＝f[x ₀ ]+f[x ₀ ,x ₁ ](xx ₀ )+ primary interpolation

f[x ₀ ,x ₁ ,x ₂ ](x-x0)(x-x1)+...+ quadratic interpolation

f[x ₀ ,x ₁ ,...x _n ](xx ₀ )(xx ₁ )...(xx _n-1 )

Rn(x)＝f[x,x ₀ ,...,x _n ]·π _n (x) interpolation remainder

When determining the zero-crossing point, the _y value is actually known, and x is calculated. In order to facilitate the use of the interpolation algorithm, all the x and _y in the formula are reversed and used, thus the three-time Newton interpolation algorithm is obtained as follows:

x=N _n (y)

=f(y ₀ )+f[y ₀ ,y ₁ ](y-y0)+f[y ₀ ,y ₁ ,y ₂ ](yy ₀ )(yy ₁ )+

f[y ₀ ，y ₁ ，y ₂ ，y ₃ ](yy ₀ )(yy ₁ )(yy ₂ )

in,

In addition, for the zero-crossing point, its y=0, thus:

In the above formula, x ₀ ＝f(y ₀ ), and (x ₀ , y ₀ ), (x ₁ , y ₁ ) are two points before the zero-crossing point; (x ₂ , y ₂ ), (x ₃ , y ₃ ) are two points after the zero-crossing point, and four points are adjacent sampling points, therefore, x ₀ -x ₁ =x ₁ -x ₂ =x ₂ -x ₃ =-1, the above formula can be written as:

Among the variables involved in the above formula, x ₀ is the time of the sampling point, and y ₀ ~ ₃ are the values of the time-domain sampling points of the adjacent four pulse signals. These parameters are all known quantities. According to the above formula, the values of each zero-crossing point can be obtained. data. After the zero-crossing data is obtained, the precise time interval between two adjacent teeth can be obtained.

2) The method of turning the in-cylinder pressure from the time domain to the angle domain

Through the above interpolation algorithm, the precise moment of the zero-crossing point of the horn signal can be obtained, that is, Tstart and Tend shown in Figure 4 below, and the duration between these two moments is the duration corresponding to one tooth of the engine flywheel ring gear. Corresponding angle θ=360°CA/number of teeth of flywheel ring gear. In this method, it is assumed that the rotational speed of the engine in one tooth is the same, so the data corresponding to any angle between the two teeth can be further obtained through the interpolation algorithm, as shown in Figure 4, which is the time domain rotation angle domain of the pressure signal in the cylinder Schematic diagram, using this diagram as an example to illustrate the implementation process. T ₀ , T ₁ , T ₂ and T ₃ in the figure are the data at each time with a difference of T _step time, and the expression of T _step is as follows:

In the formula, θ is the angle between two adjacent teeth of the flywheel ring gear; the _{interpolation interval} of θ is the angle interval transformed into the angle domain; T _start and T _end are the zero-crossing point data obtained by interpolation; each point converted into the angle domain corresponds to The expression of the moment T _n (n=0, 1, 2, 3...) is:

T _n = T start + n × T _step + T _off

In the formula, _Toff is the angle deviation between the first angle domain point obtained by the interpolation value of the current tooth and the starting time Tstart of the current tooth. For the calculation of the first tooth, _Toff is 0, and for the second one in the figure For teeth, T _off =T ₃ -Tend.

Once the time T _n (n=0, 1, 2, 3...) of the in-cylinder pressure converted into the angle domain is obtained, the value y _n (n=0, 1, 2, 3...) It can be obtained by linear interpolation of the time-domain in-cylinder pressure data (before T _{n, before} y _n ) and (after T _{n, after} y _n ) collected before and after this point, where, before T _{n and before} T _n The _latter is the sampling time of the two points before and after, and the difference between the two is 1. _Before and _{after y n} is the pressure value of the two points before and after y n. Taking T ₁ time as an example, the corresponding in-cylinder pressure value of this time is The expression for _y1 is as follows:

By adopting this method, the time-domain data collected by the internal clock acquisition card can be converted into angle-domain data whose interpolation interval is θ _{interpolation interval} °CA.

Through the above method, the time domain pressure signal in the current tooth can be transformed into the angle domain, and the above method can be used for processing in each tooth.

This method needs to use the interpolation algorithm to obtain the time of the zero crossing, and there must be corresponding points before and after the zero crossing to ensure the smooth progress of the interpolation process. The number of points is related to the interpolation algorithm used, the sampling frequency and the speed of the engine. During the working process of the engine, the rotational speed varies in a wide range. The higher the rotational speed, the shorter the cycle time. In order to ensure that the number of points required for interpolation is sufficient, the higher the sampling frequency is required, it is only necessary to ensure that the sampling frequency can obtain sufficient sampling at high rotational speeds. Points, no problem at low revs.

Different interpolation algorithms require different points before and after the zero-crossing point. The following is still explained with the third-order Newton interpolation algorithm. From the above analysis, it can be seen that when using the three-time Newton interpolation method to calculate the zero-crossing point, two sampling points are required before and after the zero-crossing point for calculation. Therefore, within the duration of one tooth, at least 8 sampling points should be collected. Temporarily calculate with 8 points. Assuming that the number of teeth of the engine flywheel ring gear is z, the number of sampling points in one cycle of the engine is 8×2×z, and the time of one cycle when the speed is n ₁ is seconds, so the sampling frequency at the current speed can be obtained as:

Assuming that the number of ring gears of the engine is 141 and the maximum operating speed is 2200r/min, according to the above formula, the sampling frequency of a single channel cannot be lower than 41.36kHz.

When collecting and analyzing the pressure signal in the cylinder, there are requirements for the number of sampling points. For example, diesel engines generally require more than 100 cycles per sampling; for gasoline engines, the number of sampling cycles is in the range of 150 or more. The sampling frequency can be determined by the above method. When the engine speed is different, the time corresponding to each cycle is different, so the number of sampling points required is also different. The lower the speed, the longer the cycle time, and the more corresponding sampling points. Analyze the calculation method of the number of sampling points, assuming that the engine speed is n ₂ and the number of cycles to be sampled is m, the expression of the corresponding number of sampling points is as follows:

Assume that the number of engine ring gears is 141, the maximum operating speed is 2200r/min, and the idle speed is 600r/min. At this time, 100 cycles of data need to be collected, and the number of sampling points required for each sampling at this time is: 827,200 points .

When the method proposed by the present invention is used in practice, the sampling frequency and the number of sampling points are reasonably selected according to the number of flywheel teeth of the engine and the operating speed range, and the in-cylinder pressure signal collected by the internal clock can be converted into an angle domain signal according to the above interpolation algorithm. for subsequent analysis.

When the present invention is applied, accompanying drawing 1 is a schematic diagram of the hardware structure of the test method proposed by the present invention. This method needs to use a cylinder pressure sensor and a pulse signal sensor, and each signal is sent into the internal clock data acquisition system as an analog signal to realize data collection. collection. Due to the large amount of collected data, the result of analog-to-digital conversion is transmitted to the computer through a high-speed data transmission interface such as USB or PCI for subsequent analysis. The internal clock acquisition system uses the internal clock for data acquisition, so the time interval of each sampling point of the signal is fixed, that is, the sampled signal is based on time. When the internal combustion engine is working, the crankshaft angle of 720 degrees is used as a cycle. When analyzing the pressure signal in the cylinder of the internal combustion engine, it needs to use the crankshaft angle as the reference for analysis. Therefore, it is necessary to convert the signal in the time domain into an angle domain signal. This process That is the content that the present invention mainly relates to.

Accompanying drawing 2 is the comparison curve after using this method to convert the measured engine cylinder pressure signal from the time domain to the angle domain. Figure 2(a) is the comparison curve of the pressure signal and pulse signal in the cylinder measured for 5 cycles, where the abscissa is the number of sampling points. Since the sampling frequency of the internal clock sampling is fixed, the number of sampling points corresponds to the time. It can be seen from the figure that the sampling frequency is high, and the number of sampling points of the pressure signal in the cylinder and the pulse signal are large. Figure 2(b) is the time-domain in-cylinder pressure signal converted into an angle-domain signal by the interpolation algorithm proposed by the present invention, and the angle interval θ _{interpolation interval} during conversion is set to 0.5°CA, that is, there are 1440 points in each cycle . It can be seen from the comparison of the two figures that the method proposed by the present invention can accurately convert the signal collected by the internal clock into the angle domain.

Although the specific implementation of the present invention has been described above in conjunction with the accompanying drawings, it does not limit the protection scope of the present invention. Those skilled in the art should understand that on the basis of the technical solution of the present invention, those skilled in the art do not need to pay creative work Various modifications or variations that can be made are still within the protection scope of the present invention.

Claims (5)

1. A method for converting a time domain of an in-cylinder pressure signal of an internal combustion engine into an angle domain is characterized by comprising the following steps:

obtaining the points of 0 output voltage of the pulse signal measuring device, namely the moments Tstart and Tend of the zero crossing point through an interpolation algorithm;

the duration time corresponding to one tooth of the engine flywheel gear ring is between the two moments Tstart and Tend, and the angle corresponding to the two moments Tstart and Tend is the duration angle theta of one tooth;

according to Tstart and Tend, the continuous angle theta of one tooth andthe angular interval theta of the in-cylinder pressure signal after converting from time domain to angular domain_{Interval of difference}Determining time step length T between adjacent points of in-cylinder pressure signal in angle domain_step；

According to the sampling interval T_stepAnd the angle deviation T between the first angle domain point obtained by the interpolation value of the current tooth and the starting time Tstart of the current tooth_offAnd obtaining the time T corresponding to the angle domain by the current tooth starting time Tstart_n，n＝0,1,2,3…；

According to the obtained angle domain corresponding time T_nN is 0,1,2,3 …, and the value y of the in-cylinder pressure at each point_nThe measured in-cylinder pressure signals at two points before and after each point are linearly interpolated, and n is 0,1,2,3 ….

2. The method according to claim 1, wherein when the output voltage of the pulse signal measuring device is 0, that is, the zero crossing point, is obtained by the interpolation algorithm, two points before and after the zero crossing point are obtained by the Newton interpolation method for 3 times, and (x) the time domain of the in-cylinder pressure signal is converted into the angle domain₀，y₀)、(x₁，y₁)、(x₂，y₂) And (x)₃，y₃) Wherein x is₀～x₃The sampling sequence number of each sampling point is, because four points are continuous sampling points, the interval of adjacent points is different by 1 sampling point moment; and y is₀～y₃Then the sampling results of two points before and after the zero crossing point are obtained, then all x and y in the Newton interpolation formula are used in a reversed mode, y is made equal to zero, and the corresponding x value is obtained.

3. The method of claim 1, wherein the angle θ between the times Tstart and Tend is the following angle θ: θ equals 360 ° CA/number of teeth of the flywheel ring gear.

4. The method of claim 1 wherein the time domain to angle domain of the in-cylinder pressure signal of the internal combustion engine,

interval time difference of T_step：

In the formula, theta is a continuous angle of one tooth of the flywheel ring; theta_{Interpolation interval}Setting an angle interval for converting into an angle domain; t is_startAnd T_endIs zero crossing data obtained by interpolation.

5. The method of claim 4 wherein the time domain of the in-cylinder pressure signal of the internal combustion engine is converted to the angular domain at a time T corresponding to each point in the angular domain_nThe expression of (a) is:

T_n＝T_start+n×T_step+T_off

wherein n is 0,1,2,3 …, T_offThe angular deviation between the first angular domain point obtained for the interpolated value of the current tooth and the starting time Tstart of the current tooth.