Target Force Curve Searching Method for Axial Electromagnetic Dynamic Balance of Scroll Compressor

Abstract

In order to explore the axial electromagnetic dynamic balance force curve of a scroll compressor under the action of thermal energy, radial clearance, mechanism friction, and other factors in actual working conditions, an axial force exploration method that can automatically approach sections is proposed in this paper. Considering the dynamic response ability of the electromagnetic balance system, an automatic optimization algorithm of the partition number was proposed to find the optimal partition number in order to achieve the optimal tracking effect. An experimental platform was built to test the effect of the segmented tracking method on calibrating the deviation between the theoretical axial force curve and the real curve. The results show that the curve construction method proposed in this paper has convergence. This method can automatically and accurately construct the axial balance force curve required by the electromagnetic dynamic balance. Through the automatic optimization algorithm, the standard error (RMSE) between the target curve and the theoretical curve was reduced from 290 to 22.6, and the number of partitions with the lowest standard error was 20. The results provide a useful reference for the accurate, automatic, and efficient exploration of the actual axial sealing force of the scroll compressor.

1. Introduction

The scroll compressor is a high-efficiency positive displacement compressor that has characteristics such as a compact structure, stable operation, and low noise. With the continuous development of manufacturing technology, the application range of scroll compressors has been expanding in multiple fields, such as refrigeration, power, dehumidification, and exhaust gas treatment [1,2,3,4]. Radial leakage through the axial clearance is a hot topic in the research field of the scroll compressor. The change in the sealing gap size between the dynamic and static scrolls have a significant impact on working efficiency and reliability. Its development and progressive technologies are based on reasonable design, processing, and effective control.

Radial leakage and tangential leakage are the main gas leakages of scroll compressors. Radial leakage occurs through the axial clearance between the tip of the involute and the opposite scroll base. This gap causes the leakage of gas from the high-pressure chambers to the low-pressure chambers, which reduces the exhaust pressure and the repeated compression of the gas, reducing the working efficiency of the scroll compressor.

The axial gas force is the sum of the gas forces in each gas chamber, and each gas force is the product of the area of each gas chamber and its air pressure. The size of the air cavity area is related to the geometric model of the compressor. Therefore, many scholars have begun to study the calculation formulas of the axial gas force of scroll compressors under different geometric models. Japanese scholar Yoshio Morita first established the geometric theory of circular involute and deduced the axial force formula of the vortex disk [5,6]. Professor E.A. Groll of Purdue University, etc., comprehensively established the mathematical model of the rotary compressor in the processes of suction, compression, and exhaust, and analyzed the thermal characteristics, dynamic characteristics, and pool leakage chemistry of the rotary compressor in detail [7]. Yanagisawa et al. devised the general expression of the volume of all vortex chambers and established a complete compressor analysis model consistent with the experiment [8]. In [9], a novel method to describe the geometry of scroll teeth using a new reference system was proposed, and the thermodynamic model of a scroll compressor was established; in [10], the influence of different scroll profiles on the performance of scroll compressors was studied. Many scholars have studied the modeling of the scroll compressor in order to analyze its internal dynamics, its thermodynamics, etc. [11,12,13,14].

The normal solutions for the axial sealing of scrolls include the following: (1) Open a back pressure hole on the orbiting scroll and introduce the gas in the compression cavity into the back pressure cavity to provide back pressure to balance the axial direction of the orbiting scroll [15,16,17,18]. Although the gas force and the gas back pressure scheme can offset a large portion of axial pressure, it is extremely difficult to match the dynamic performance when the scroll machine is running at high speed, which leads to the weakening of the sealing effect. (2) Use the oil circuit to introduce high-pressure lubricating oil into the back pressure chamber to provide a constant back pressure to balance the axial gas force on the orbiting scroll [19,20]; it is difficult to adjust the balance problem between the gas leak and mechanical wear of the vortex disk in the scroll compressor. (3) Use the limit structure to limit the axial gap between the moving and static scrolls and reduce leakage [21,22]. As the axial position is fixed, the automatic compensation of the axial gap cannot be realized, and it is difficult to seal a small balance with pressure. (4) Install a sealing strip on the end face of the scroll, and open a labyrinth groove structure on the top of the scroll tooth [23,24,25].

Based on the theory of scroll compressors, in [26], an oil-free scroll compressor was designed using magnetic drive technology. In [27], a scheme was proposed for dynamically balancing the axial gas force of the scroll compressor by using electromagnetic force. The scheme makes use of the magnetic characteristics of electromagnets, which can be changed by controlling the current, and designs an electromagnetic mechanism with a simple structure. The mechanism can generate dynamic electromagnetic force through the PID control method, which is used to solve the problem of radial leakage during the operation of the scroll machine and realize oil-free sealing. In [28], a full-frequency range adaptive electromagnetic force dynamic tracking algorithm was proposed and verified by hardware in the loop experiment, showing that the algorithm can effectively adapt to the working conditions of scroll machines under different speed conditions.

The axial separation force target curve of the scroll compressor mentioned in [27,28] is the axial gas force curve estimated by the scroll model theory, and the tangential inter-leakage, gap leakage between structures, heat energy, and friction between transmission mechanisms during the actual operation of the scroll machine, affects the air pressure and its geometric area in the air pressure chamber. Thus, the calculated axial gas force theoretically deviates from the actual force [29,30,31,32,33].

In this paper, an automatic piecewise approximation tracking method, which is on the basis of [27], is proposed. The proposed method is to automatically find the required axial balance force curve during the operation of the compressor. Then, considering the effects of various factors on the tracking effect, the single parameter of the largest number of zones was selected to optimize. An automatic optimization control method of the tracking partition number was proposed for making an automatic piecewise approximation tracking method work better. Finally, an experimental platform was built for verification.

The experimental results show that the probe scheme has little dependence on the mechanical parameters of the scroll compressor. By controlling the magnitude of the electromagnetic force, the exhaust volume and current of the scroll compressor can be controlled and monitored in real time to the optimal values, and the axial balance force target curve can be automatically adjusted to the optimal curve to realize the optimal control of the axial separation force of the scroll compressor balanced by the electromagnetic force. This provides power for the further development of subsequent scroll compressors.

2. Electromagnetic Dynamic Balance Control for Scroll Compressors’ Axial Separation Force

2.1. Mechanical Structure Diagram of the Electromagnetic Balance Mechanism

According to the relation between electromagnetic suction and current, a balancing scheme of the axial separation force of the compressor was designed based on [27]; Figure 1a shows the top view of the structure, and Figure 1b shows the sectional view from the A-A direction. Figure 1b shows the force diagram of the scroll disc. The axial gas force Fa exerted by the scroll disc causes the scroll disc to disengage along the axis, leading to the leakage of compressed gas and the reduction in the volume efficiency of the compressor, which is one of the main defects of the scroll compressor.

The relationship between electromagnetic suction and current in an electromagnetic mechanism can be expressed by the following formula:

$$F_c\left(i,z\right)=\frac{A\mu N_1{}^2}{4}{\left(\frac{i\left(t\right)}{z\left(t\right)}\right)}^2$$

(1)

where A represents the surface area through which the magnetic line of force passes vertically, μ is the air permeability, N₁ indicates the number of turns of the electromagnetic coil, i(t) describes the current through the electromagnetic coil, and z(t) is the air gap distance between the core and the armature.

From the formula, it can be seen that the electromagnetic suction is proportional to the square of the current and inversely proportional to the square of the air gap.

2.2. Axial Gas Separation Force

The Fa calculation formula of the axial gas force on the swirl disc [6] is as follows:

$$F_a=\left\{\begin{array}{c}\pi P_s{P}_t^2\left[\frac{A_1}{\pi P_t^2}{\rho }_1+{\displaystyle {\displaystyle \sum }_{i=2}^{N_1}}\left(2i-1-\frac{\theta }{\pi }\right){\rho }_i\right]\begin{array}{cc}& 0°<\theta <{\theta }^{*}\end{array}\\ \pi P_s{P}_t^2\left[\frac{A_1}{\pi P_t^2}{\rho }_1+{\displaystyle {\displaystyle \sum }_{i=3}^{N_1}}\left(2i-1-\frac{\theta }{\pi }\right){\rho }_i\right]\begin{array}{cc}& {\theta }^{*}<\theta <{360}^{°}\end{array}\end{array}\right.$$

(2)

where P_s is the suction pressure, P_t is the pitch of the involute, A₁ shows the action area of the axial gas force in the central compression cavity, N₁ is the number of cycles of the involute, ρ_i describes the pressure ratio of the first compression chamber, and ϴ* shows the corresponding angle of different exhaust pressures.

By substituting the model prototype parameters into the calculation, the relationship between the axial gas force and the spindle rotation angle can be obtained as shown in Figure 2 below.

It can be seen from the above diagram that the axial gas separation force of the scroll machine changes greatly and periodically with the rotation angle of the main shaft.

For such axial gas forces with large periodic variation, common gas back-pressure balancing schemes and limit structure balancing schemes can balance the axial gas forces to a certain extent, but it is difficult to provide the axial balancing force required for a scroll compressor under high exhaust pressure. The electromagnetic balance scheme takes the axial gas separation force as the tracking target and feeds back the electromagnetic suction generated by the electromagnetic mechanism with the iron core through the force sensor. Meanwhile, the electromagnetic suction (F_c) is dynamically adjusted by changing the current of the solenoid coil in real time by using the PID controller to achieve the scheme of balancing the axial gas separation force (F_a).

As there is phase lag between periodically varying the current and electromagnetic force in the electromagnetic control system, the phase advanced compensation algorithm is proposed by the authors to solve the error caused by tracking lag. The research results show that the target curve can be tracked quickly under low-frequency rotation of the main shaft, and the axial separation force can be well-balanced.

2.3. Sealing Effect with Theoretical Axial Gas Force as the Target Force

The frequency of the spindle motor is 25 Hz. With the theoretical axial gas force curve at 5 bar exhaust pressure as the tracking target curve, the scroll compressor is sealed axially by the above electromagnetic balance mechanism. At this time, the tracking state of electromagnetic suction is shown in Figure 3 below.

As can be seen in Figure 3 above, the electromagnetic suction can effectively track the theoretically calculated axial gas force curve. Under the condition of good electromagnetic suction tracking, the volume efficiency of the scroll compressor in the experiment was smaller than the expected volume efficiency, so the compression efficiency of the scroll compressor in this system has room for improvement.

In order to solve the above problems, this paper proposes a piecewise approximation tracking method to directly study the tracking of the scroll compressor in the target state, which can automatically, quickly, and accurately obtain the actual axial target force curve.

3. Target Force Curve Searching Method

3.1. Evaluation Criteria of the Target Force Curve Searching

The electromagnetic force-tracking objects proposed in Section 2 are all theoretically calculated axial gas force curves, while the electromagnetic seal mainly aims at balancing the axial gas force of the scroll machine and the force acting between the drive. Therefore, a section approximation tracking method is proposed in this paper, which can accurately, automatically, and efficiently measure and construct the required dynamic axial balance force when considering the dynamic response of the electromagnetic system. The influence of interval on the relevant parameters of the sectional approximation tracing method is discussed, and an automatic zoning optimization algorithm is proposed to achieve the optimal tracing effect. Exhaust flow is an important index to check the compression efficiency of the scroll machine. Therefore, in a unit time, the exhaust flow generated by the scroll machine is within the expected range, which means that the compressor gas leakage is small and its axial seal is good. However, if the electromagnetic force is too large, and the moving and stationary scroll discs are close to each other, the friction loss between the two will increase the torque of the spindle motor, and the current will increase accordingly. Therefore, in the method of searching the axial target force curve, the exhaust flow value and the current value of the spindle motor are the main indicators to evaluate the effectiveness of the axial target force calibration.

3.2. Automatic Segmented Control Method for the Target Force Curve Searching

The axial separation force in other literature refers to the axial gas load [9,10,11]. The axial gas load is produced by the collection of gas pressures acting in the respective sealed chamber, which can be calculated by:

$$F_a\left(\theta \right)={\displaystyle \sum }_{i=1}^{N_1}{p}_i{A}_i\left(\theta \right)\hspace{1em}\left(0\le \theta \le 2\pi \right)$$

(3)

where

$$p_i=P_s{\left[\frac{A_1\left(0\right)}{A_i\left(\theta \right)}\right]}^k\hspace{1em}\left(0\le \theta \le 2\pi \right)$$

(4)

where P_s indicates the suction pressure of the scroll compressor, and k represents the adiabatic index.

The solution methods of the compression cavity under different involutes are introduced in [10,11]. However, in the actual working process of the compressor, the volume of the compression chamber is affected by many factors; coupled with the axial force of the transmission mechanism on the scroll plate, there is a deviation between the actual axial separation force curve and the original axial gas separation force curve, which increases the difficulty of constructing the mathematical model of the scroll compressor. This paper proposes the automatic segmented control method for the target force curve searching.

The black curve in Figure 4 represents the axial separation force curve constructed based on the theoretical model, and the red curve represents the axial separation force curve under actual working conditions.

The actual axial separation target force curve is unknown in the actual compressor working process, so this paper uses the concept of integral to calibrate the theoretical axial target force curve. Its main working principle is shown in Figure 4, Figure 5, Figure 6 and Figure 7 below: the black curve in the figure is the hypothetical actual axial separation target force curve. First, the red curve is divided into N equal angle intervals according to a cycle of 2𝜌 rotations of the main shaft, and the width of each angle area is 2𝜌/N, as shown in Figure 4.

Next, for the first interval 0–(m = 2𝜌/N), the corresponding axial forces are all set to the value at the 0° of the theoretical axial gas force curve, as shown in Figure 5a.

Then, we use the phase-advance-PI algorithm to make the electromagnetic force keep up with the “actual curve” and adjust the interval electromagnetic force until the exhaust volume and current in the interval reach the optimal state. At this time, the interval is calibrated as shown in Figure 5b. Finally, the next N-1 angle areas are calibrated according to the above method until the entire curve is calibrated. The curve after calibration is shown in Figure 6.

According to the above tracking method, the control flow chart shown in Figure 7 was obtained.

First, judge whether the actual axial separation force is greater than the theoretical axial balance force by increasing the axial target force. If so, the upward exploration process will be carried out, as shown in Figure 8a; otherwise, the downward exploration process will be carried out, as shown in Figure 8b.

In this paper, the electromagnetic balance mechanism combined with the optimization algorithm is used to automatically and effectively find the axial balance force curve with a good sealing effect. The original curve, such as the black curve in Figure 4, can be obtained through the simplified mathematical model of the scroll compressor.

4. Influence of System Parameters on Response

4.1. Influence of Partition Number on the Tracking Effect

It can be seen from Section 3.2 that the probe tracking method is based on the concept of integration to find the target curve. When the number of partitions n is more—that is, the shorter the length of the integrator interval, and the smaller the area enclosed between the two curves in the same interval—it means that the closer the curve obtained by the piecewise approximation tracking method is to the real target curve, the smaller the axial leakage of the scroll compressor, as shown in Figure 9a,b.

It can be seen that with the increase in the number of partitions in Figure 9b, the target curve obtained by tracking is obviously closer to the real target curve than that in Figure 9a.

However, the electromagnet energizing circuit is similar to the RL circuit. Due to the existence of inductance, the electromagnet energizing current cannot change instantaneously with the voltage. Due to the charging and discharging time, the current needs to adjust the time from one stage value to another. According to the relationship between the electromagnetic force and current, it shows that there is a similar situation of electromagnetic suction when tracking the axial separation force of the target. The tracking step signal can reflect the dynamic response ability of the electromagnetic suction under the system; that is, the shorter the time spent from a fixed value to another fixed value, the stronger the dynamic response ability. Figure 10 shows the dynamic response capability of the current system.

In Figure 10, the frequency of the target curve is 5 Hz, and the tracking target curve is a square wave with a maximum value of 1500 N and a minimum value of 1000 N. At this time, the adjustment time of the electromagnetic suction rise is 16 ms; the adjustment time of the electromagnetic force drop is 23 ms.

If the response of the electromagnetic balance mechanism system is fast enough, the electromagnetic force output curve will be controlled into a stepped curve consistent with the target curve, such as the red curve in Figure 11a,b. However, due to the system response capability, after the number of zones reaches a certain range, the electromagnetic force cannot be controlled into a ladder curve consistent with the target curve, as shown in Figure 11b. The current signal frequency is 25 Hz.

Comparing Figure 11a with Figure 11b, it can be seen that when the number of zones is 5, the electromagnetic force output energy follows the target curve in a stepped distribution; when the number of zones is 20, because the time corresponding to each zone decreases, the force of the electromagnetic force in the current system changing in one zone will decrease, and the target force will be reached only within the specified time or not at all.

At that time, in the tracking process, if there are too many divided angle intervals, and the time interval between adjacent angle areas is too small, the electromagnetic force will track the next stage when it is not stable, resulting in the phenomenon in which the tracking results are not improved, but occupy the calculation time of the controller.

4.2. Automatic Optimization Algorithm of the Partition Number

Recently, there have been many good literatures on parameter optimization [34,35,36,37], such as [34,35]. Considering the amount of calculation of the optimization algorithm and only the single parameter of partition number, the optimization was carried out. In this paper, an automatic optimization control method of the tracking partition number based on piecewise approximation is proposed.

The main principle of the control method is that the value of the probe tracking partition number n is continuously optimized iteratively in the set threshold m (360), and the exhaust flow and current of the scroll compressor are used as the evaluation basis. When the exhaust flow of the scroll compressor reaches more than 85%, on this basis, the minimum current is the optimal state, the value of the number of probe tracking zones n that is most suitable for the system can be determined, and the N value can be saved for a direct call in the next tracking. The flow chart of the segmented approximation tracking partition number automatic optimization control method is shown in Figure 12 below.

5. Experiment and Results

5.1. Experimental Platform

In order to verify the theoretical analysis and the proposed control method, a probe tracking simulation experimental platform based on the electromagnetic dynamic balance mechanism was built. The experimental platform is shown in Figure 13, and its parameters are shown in

Table 1. Experiment platform.

Name of Parameter	Parameter Values
Turns of solenoid coils	200
Diameter of solenoid coil air	100 mm
Air gap of iron core	0.3 mm
Revolver pair series	1 pole-pair
Pressure sensor range	0~2000 N

.

The experimental platform software compilers were all Keil5 compilers. Various data were communicated and transmitted with the host computer through the serial port, and the corresponding parameter change curve and oscilloscope display were drawn.

5.2. The Steps of the Experiment

In order to prove the feasibility of the above-mentioned tracking method, we built a simulation experimental platform. As the scroll prototype was lacking, the exhaust flow and current value could not be used for determining the calibration of the axial gas separation force. Therefore, as the tracking method remained unchanged, an axial force curve that was not much different from the theoretical curve was proposed as the actual axial separation force curve, and the variance between the calibration curve and the true curve was used as a criterion for this experiment. The calculation formula for the sum variance mentioned is:

$$\begin{array}{cc}{R}_e={\displaystyle {\displaystyle \sum }_{\alpha =N\ast m}^{\left(N+1\right)m}}{\left(F_{\alpha }-f_{\alpha }\right)}^2& \left(N=0,1,2,\cdots \cdots ,\frac{360}{m}-1\right)\end{array}$$

(5)

where R_e describes the effect of current angle area curve tracking, N represents the number of probe trace zones, m is the angle in each angle area, F_α is the electromagnetic force at the target angle, and f_α is the axial target separation force at the target angle.

Therefore, the main difference between the semi-physical experiment control flow chart and the physical experiment was the judgment module:

Change whether the exhaust flow increases to whether the sum variance between the two curves in the current calibration zone increases, and remove the judgment of the spindle drive motor current.

The search for the optimal number of zones was carried out by means of the sectional automatic optimizing control method for the traceable zoning number. In this paper, the RMSE (root mean square error) is proposed as the criterion for the optimum zoning number optimizing control method to ensure the matching between the electromagnetic force output curve and tracking curve. The calculation formula is:

$$Err=\sqrt{\frac{{{\displaystyle \sum }}_{i=0}^{359}{\left(F_i-f_i\right)}^2}{360}}$$

(6)

where Err shows the effect of curve tracking, F_i is the electromagnetic force at the target angle, and f_i is the axial target separation force at the target angle.

When the spindle frequency is 25 Hz, manually change the number of zones, and then carry out the piecewise approximation tracking experiment; the experimental results are shown in

Table 2. Tracking error change table under each time zone.

The Number of Probe Tracing Zones—N	RMSE
10	61.01
15	27.85
40	22.89
72	25.16
180	25.81
360	25.94

below.

It can be seen from the table that when the number of zones was small, the root mean square error between the electromagnetic force output curve and the proposed real target curve decreased with the increase in the number of zones. After the number of zones was 15, the error change tended to be stable.

In order to find the best number of zones in the current system, the automatic optimization algorithm of traceable sectional zones was partially modified.

The flow chart of the semi-physical sectional automatic optimization algorithm for the number of probe trace zones is shown in the following Figure 14.

5.3. The Results of the Experiment

Figure 15 shows the experimental results obtained using the segmented probe tracking method.

Figure 15a shows the theoretical curve and the real axial force curve before the segmented tracking experiment. It can be seen from the figure that there is a big gap between the two curves. The RMSE between the two curves was 290. Figure 15b is the calibrated experimental curve. It can be clearly seen from the figure that the electromagnetic force output curve and the actual axial force curve closely fit. When the number of probe trace zones was 20, the RMSE between the two curves was 22.6.

Figure 16 shows the variance in the first interval with the number of times adjusting the calibrated axial force curve during the segmented tracking process. It can be seen from Figure 16 that the standard error first became smaller and then larger. During this process, there was a minimum. This shows that the segmented probe tracking method has convergence.

Figure 17 is a graph showing the variation of the error between the electromagnetic force output curve obtained by the automatic optimization algorithm and the actual curve with the number of probe trace zones. It can be seen that, after the number of probe trace zones reached 15, the tracking effect of the electromagnetic force was relatively stable. When the number of probe trace zones was 20, the tracking error was the smallest, which means that, under the current working conditions, the tracking effect of the electromagnetic force is the best.

6. Conclusions

This paper studies the construction method of the axial separation force curve of the scroll compressor and optimizes the parameters of a partition number in this method.

First, in order to solve the problem of deviation between the theoretical curve and the actual curve, an automatic real-time tracing scheme was proposed to realize the automatic calibration of the theoretical curve under the condition of 7 bar exhaust pressure at 25 Hz frequency. Compared with the traditional method of constructing the axial separation force curve, the requirement for accuracy of the mathematical model of the scroll compressor was reduced.

Second, the influence of the partition number on the tracking effect was studied. Under the current operating conditions, the tracking effect was obvious when the number of time zones increased in the range of 1–20. After the number of time zones was 20, the tracking effect did not improve; on the contrary, it slightly deteriorated when the number of time zones increased.

Finally, an electromagnetic balance experimental platform was built and tested to verify the feasibility of the sectional approximation control method and the optimization algorithm. The experimental results show that the automatic piecewise approximation tracing method in this paper has good convergence, and each time zone can track the target curve by iteratively modifying the electromagnetic force until the axial sealing effect is the best, and the optimization algorithm can effectively improve the curve tracing result.

The method provided in this paper can search for the true axial force separation curve under corresponding operating conditions, thus improving the effect of the electromagnetic force-tracking curve, and provide important technical support for subsequent electromagnetic force axial sealing.