A new dynamic displacement prediction method for contactors with constant air gap

Abstract

In order to improve the dynamic characteristics of the contactor, this paper proposes a new dynamic displacement prediction method for contactors with constant air gap. The dynamic displacement process can be predicted by observing the flux linkage produced by the permanent magnet. The equivalent magnetic circuit model of electromagnetic force driving actuator (EMFA) is established. The formulas of dynamic flux linkage, displacement and current are deduced, and the displacement prediction of the contactor is completed. The curved surfaces of driving force, flux linkage, displacement and current are established by finite element method. The movement process of the contactor is dynamically simulated by interpolation method. Finally, an EMFA-driven contactor is used for the experiment. Experimental results show that the new dynamic displacement prediction method is suitable for contactors with constant air gap. On the basis of sensorless control, bounce time of the contact can be effectively reduced by adjusting the PWM duty cycle and the corresponding displacement control parameters. The research in this article can be used to guide the design of future controllers. And the simulation results are consistent with the experimental results. The experiment verifies the effectiveness of the new dynamic displacement prediction method.

Keywords

Contactor actuator constant air gap sensorless

1. Introduction

As a new type of actuator, electromagnetic force driving actuator (EMFA) is gradually applied to power switching equipment of various voltage levels [1–4]. In addition to high-voltage circuit breakers, an improved EMFA is used in contactors due to its constant air gap and short excitation time [5]. The driving force of the constant air gap actuator will not increase rapidly due to the reduction of the air gap. This is conducive to the suppression of contact bounce. With the development of power distribution systems, higher requirements are put forward for the reliability and life of low-voltage contactors. Therefore, it has certain engineering significance to carry out related research on contactors with constant air gap.

The control and modeling methods of electromagnetic equipment have been studied in depth [6–11]. In [12], a contactor control method based on artificial neural networks is proposed. In [13], a technique for controlling the coil current by PWM chopping is proposed. Current and displacement sensors are used to track the vacuum circuit breaker to make it act according to a given curve [14]. A flux-weakening control method is proposed to improve the dynamic characteristics of the actuator [15]. The permanent magnetic contactor is studied, and the movement position of the contactor is calculated by observing its dynamic inductance [16]. In [17], a displacement sensorless control method based on fuzzy control is proposed.

At present, EMFA is mainly used in power switches, such as circuit breakers, relays, contactors and load switches. In addition, as an actuator, EMFA has the ability to convert electrical energy and mechanical energy. Therefore, it also has certain feasibility in the application of vibration control [18,19].

Research on modeling and control of variable air gap contactors has achieved remarkable results[20–24]. The inductance of the variable air gap contactor changes with the dynamic position. When the air gap gradually decreases, the magnetic resistance of the magnetic circuit gradually decreases, and the inductance gradually increases. The dynamic position can be predicted by dynamic inductance [17]. However, this method is not suitable for contactors with constant air gap. Therefore, it is necessary to analyze the model and find a new type of displacement observation method.

In view of the above problems, this paper derives the calculation formula of the EMFA dynamic displacement based on the equivalent magnetic circuit method. The new control method can predict the dynamic displacement process of the actuator through the flux linkage generated by the permanent magnet. Then the surface of driving force, flux linkage, displacement and current are established by finite element method, and the simulation of the dynamic process is completed. Finally, experiments are carried out for different PWM duty ratios and displacement control parameters. The simulation results are consistent with the experimental results.

2. The basic structure of EMFA

The contactor designed in this article is driven by EMFA, and its schematic diagram is shown in Fig. 1. Its main structure includes: top cover, yoke, static iron core, permanent magnet, moving iron core and coil. In terms of steady state, the permanent magnet generates a magnetic field. In the transient state, the coil is discharged by the capacitor, and the current passes through the magnetic field vertically. The electrons in the coil are affected by the Lorentz force to generate driving force, driving the EMFA to move. The size of EMFA is less than 40 ∗ 40 ∗ 40 mm³. The ferromagnetic material of the contactor is DT4E, the permanent magnet material is N35, the remanence of the permanent magnet is 1.1 T, and the contact material is AgSnO2.

Fig. 1.

Schematic diagram of EMFA.

3. Mathematical model of EMFA

3.1. Magnetic circuit model

The permanent magnet of EMFA is composed of four permanent magnets, and the permanent magnets become the excitation source of the magnetic circuit. The four permanent magnets not only provide the holding force, but also provide the magnetic field for transient motion. The magnetizing directions of the two permanent magnets on the left and the two permanent magnets on the right are opposite, and the magnetizing directions of the permanent magnets on the same side are the same.

The EMFA equivalent magnetic circuit model in the motion phase is established through the above model. The magnetoresistance distribution of each part is shown in the Fig. 2.

Fig. 2.

Magnetic circuit diagram of EMFA.

3.2. Dynamic characteristics analysis

In terms of dynamic characteristics, the energized coil is subjected to ampere force in the constant magnetic field provided by the permanent magnet. During the movement of the EMFA, the air gap remains unchanged and the coil inductance changes less. The magnetic flux 𝜙 of the main magnetic circuit is an important parameter for transient calculation. In Fig. 2, the EMFA main magnetic circuit is composed of four branches. According to the superposition theorem, the four magnetic circuits are divided and calculated. Sources I ₁ N ₁, I ₂ N ₂, and magnetoresistance $\mathfrak{R}_{5}$ , $\mathfrak{R}_{12}$ , $\mathfrak{R}_{9}$ , $\mathfrak{R}_{14}$ are composed. The calculation formula of $\mathfrak{R}_{5}$ is: $\begin{eqnarray}\displaystyle \mathfrak{R}_{5}=\mathfrak{R}_{m1}+\mathfrak{R}_{m2}+\mathfrak{R}_{a}. & & \displaystyle\end{eqnarray}$ (1)

Among them, $\mathfrak{R}_{a}$ is the magnetic resistance of the air gap between the permanent magnets, and $\mathfrak{R}_{m}$ is the magnetic resistance of the permanent magnets. It can be known from the magnetic circuit rule. $\begin{eqnarray}\displaystyle I_{1}N_{1}+I_{2}N_{2}=2H_{c}h_{m}=k_{{\sigma}}{\phi}_{y}(2\mathfrak{R}_{5}+2\mathfrak{R}_{12}+\mathfrak{R}_{14}+\mathfrak{R}_{9}). & & \displaystyle\end{eqnarray}$ (2)

H _c is the coercivity; h _m is the thickness of the permanent magnet; k _σ is the magnetic flux leakage coefficient; 𝜙_y is the total magnetic flux of one of the four magnetic circuits generated by the permanent magnet in Fig. 3.

Fig. 3.

Permanent magnet flux and coil excitation flux.

Due to the parallel relationship between the moving iron core and the working air gap, the flux of the moving iron core becomes the main component of flux leakage. Regarding the permanent magnet as the distributed parameter calculation, the magnetomotive force generated by the permanent magnet per unit length is equal. Since the permeability of DT4E is much greater than the permeability of the air gap, part of the magnetic flux passes through the moving core. Therefore, the flux leakage coefficient can be approximately expressed as: $\begin{eqnarray}\displaystyle k_{{\sigma}}=\frac{d_{y}}{d_{y}-2d_{d}}. & & \displaystyle\end{eqnarray}$ (3)

According to formula (2), it can be obtained that the permanent magnet generates the magnetic flux in one of the four magnetic circuits. Among them, d _y is the height of the permanent magnet; d _d is the height of the moving iron core. The total flux linkage 𝜓 is related to the moving position x and the excitation current i. The total magnetic flux is generated by permanent magnets and coils.

As shown in Fig. 3, the solid line is the direction of the magnetic flux generated by the permanent magnet. The magnetic flux generated by the permanent magnet consists of four parts. The dotted line is the direction of the magnetic flux generated by the coil excitation, and the main path is composed of electrical pure iron.

The flux linkage generated by the permanent magnet refers to the part of the magnetic flux generated by the permanent magnet multiplied by the number of turns of the coil. As shown in Fig. 3, the flux linkage generated by the permanent magnets partially cancel each other. Taking the opening position as an example, the flux linkage generated by the permanent magnet is equal to the inverse number of the magnetic flux of the permanent magnet above the coil multiplied by the number of turns, as shown by the red line in Fig. 3. It is considered that the magnetic flux generated by the permanent magnet per unit height is the same, and the magnetic flux generated by the permanent magnet satisfies the relationship: $\begin{eqnarray}\displaystyle {\psi}_{y}=-(d_{x}-2x)\frac{4}{d_{y}}{\phi}_{y}N. & & \displaystyle\end{eqnarray}$ (4)

Where d _x is the stroke of the actuator; x is the moving position of the coil; d _y is the height of the permanent magnet; 𝜙_y is the total magnetic flux of one of the four magnetic circuits generated by the permanent magnet in Fig. 3; N is the number of coil turns, when x = 0, 𝜓_y is the initial flux linkage of the actuator. And the coil flux meets the relationship: $\begin{eqnarray}\displaystyle {\psi}_{l}\text{} & =\text{} & \displaystyle \frac{N^{2}i}{\mathfrak{R}}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \mathfrak{R}\text{} & =\text{} & \displaystyle \mathfrak{R}_{11}+\mathfrak{R}_{12}+\frac{1}{2}(\mathfrak{R}_{1}+\mathfrak{R}_{13}+\mathfrak{R}_{14}+\mathfrak{R}_{9}).\end{eqnarray}$ (6)

In order to verify the rationality of the magnetic circuit method to calculate the flux linkage, the finite element calculation results are compared with the equivalent magnetic circuit calculation results. The EMFA is fixed at a specific position and its dynamic flux linkage is calculated. The changing trend of the flux linkage is shown in Fig. 4(a). It can be seen from the curve that the flux linkage changes at different positions have the same trend. At the same time, EMFA at different locations have different magnetic linkages. This is caused by the flux linkage generated by the permanent magnets.

Fig. 4.

Flux linkage calculation of finite element and equivalent magnetic circuit. (a) Flux linkage generated by coils at different positions. (b) The relationship between displacement and flux linkage produced by permanent magnets.

As shown in Fig. 4(b), the moving position has a linear relationship with the flux linkage generated by the permanent magnet, which satisfies the derivation of Eq. (4).

In the calculation process of formula (6), the influence of magnetic saturation on the permeability of ferromagnetic materials needs to be considered. The permeability is fitted by Froelich’s equation. In the equation, a = 0.0011, b = 0.0005 [17]. $\begin{eqnarray}\displaystyle {\mu}_{rm}=\frac{a}{{\mu}_{0}(1+bH)}. & & \displaystyle\end{eqnarray}$ (7)

Then combining formulas (2), (4), (5), and (6), the dynamic displacement of EMFA can be expressed as: $\begin{eqnarray}\displaystyle x\text{} & =\text{} & \displaystyle \left(\int _{0}^{t}(U_{c}-iR)∼\text{d}t-\frac{N^{2}i}{\mathfrak{R}}\right)\frac{d_{y}}{8{\phi}_{y}N}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle \left(\int _{0}^{t}(U_{c}-iR)∼\text{d}t-\frac{N^{2}i}{\mathfrak{R}}\right)\frac{d_{y}k_{{\sigma}}(2\mathfrak{R}_{5}+2\mathfrak{R}_{12}+\mathfrak{R}_{14}+\mathfrak{R}_{9})}{16NH_{c}h_{m}}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle \left(\int _{0}^{t}(U_{c}-iR)∼\text{d}t-\frac{N^{2}i}{\mathfrak{R}_{11}+\mathfrak{R}_{12}+\frac{1}{2}(\mathfrak{R}_{1}+\mathfrak{R}_{13}+\mathfrak{R}_{14}+\mathfrak{R}_{9})}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \frac{d_{y}k_{{\sigma}}(2\mathfrak{R}_{5}+2\mathfrak{R}_{12}+\mathfrak{R}_{14}+\mathfrak{R}_{9})}{16NH_{c}h_{m}}.\end{eqnarray}$ (8)

In the above formula, the d _x component is eliminated as the initial value of the integral. Finally, the position prediction of the EMFA in the transient process can be completed by the above formula. On the basis of displacement prediction, the relevant design of the controller can be completed, which is also an important research direction of the author in the later period.

According to Eq. (8) and the experimentally measured coil voltage and current, the displacement is calculated to obtain the red dashed line in Fig. 5. The integral is calculated using Euler’s method. And the black solid line is the displacement measurement value of EMFA. As shown in Fig. 5, the difference between the measured value and the calculated value in the experiment is small, which meets the needs of no displacement sensor control.

Fig. 5.

Comparison of two dynamic displacement prediction methods.

The disadvantage of the new displacement prediction method is its application range. The total flux linkage of the variable air gap contactor has its position information, and it is not necessary to extract the flux linkage generated by the permanent magnet. The dynamic inductance method is more suitable for variable air gap contactors. The new displacement prediction method is more suitable for displacement prediction of constant air gap contactor.

Dynamic displacement prediction is one of the key technologies of displacement sensorless control. The classical displacement prediction method of contactor includes dynamic inductance method and direct calculation method [15,17,25]. Because the dynamic inductance method is not suitable for constant air gap contactors with almost constant inductance. Therefore, the direct calculation method is used to compare with the new displacement prediction method. The direct calculation method is a classical displacement prediction method. The relationship between electromagnetic force, displacement and current is obtained by finite element software. The electromagnetic force is divided by the mass and integrate it twice to get the displacement. After measuring the actual current, the dynamic displacement is solved by solving D’Alembert’s equation of motion.

Since the direct calculation method needs to calculate the electromagnetic force of the actuator, errors are introduced in the calculation process of the electromagnetic force. Therefore, the direct calculation method may produce certain errors in the later stage of the movement. The new displacement prediction method does not need to calculate the electromagnetic force to calculate the displacement, which is similar to the dynamic inductance method.

3.3. Circuit model and dynamic model

The control circuit of EMFA is shown in Fig. 6, discharges the coil by the electric capacity C. The DC power supply voltage is 72 V. The capacitance of the capacitor is 1410 μF. The chopping frequency of PWM is 10 kHZ. The circuit structure is derived from the literature [14,15]. Control methods such as [15–17] require displacement prediction as a basis. The research in this paper can provide a reference for displacement prediction of displacement sensorless control.

Fig. 6.

Control circuit.

The control system can be divided into data acquisition module, ADC module, PWM module, photocoupler module and power electronic device module. The model of DSP is TMS28335, the model of photocoupler is A316J, and the model of IGBT is G160N60.

The data acquisition module collects the voltage and current of the coil and then inputs it to the ADC module. The DSP outputs the corresponding duty cycle PWM according to the program. After the PWM enters the photocoupler, it enters the power electronic device to complete the displacement sensorless control of EMFA.

The chopping process is divided into two stages: discharge and freewheeling. The state equations of discharge and freewheeling are as follows: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle \frac{d{\psi}(t)}{dt}=\pm U_{c}(t)-Ri(t)\\[10.0pt] \displaystyle \frac{dU_{c}(t)}{dt}=\mp \frac{i(t)}{c}\\[10.0pt] \displaystyle \frac{dv(t)}{dt}=\frac{F_{1}-F_{f}}{m}\\[10.0pt] \displaystyle \frac{dx(t)}{dt}=v(t).\end{array}\right. & & \displaystyle\end{eqnarray}$ (9)

Among them, 𝜓(t) is the coil flux; U _c is the capacitor voltage; R is the coil resistance; i(t) is the current; c is the capacitance; x(t) is the movement displacement of the actuator; v(t) is the speed of movement; F ₁ is the ampere force generated by the coil; F _f is the contact force; m is the total mass of the moving part. When the capacitor discharges to the coil, U _c takes positive; i(t) takes negative. When the coil is freewheeling to the capacitor, U _c is negative; i(t) is positive.

4. Dynamic characteristics simulation and experiment

The simulation uses Maxwell to calculate the curved surface. And the dynamic characteristics of the actuator are simulated by interpolation. The experiment divides the PWM into two stages with the displacement control point. The two stages use different PWM duty ratios for speed regulation. And the two stages are distinguished by displacement control points. The displacement control point is predicted by the flux linkage generated by the permanent magnets.

4.1. Transient characteristics calculation

The parameter sweep method is used to calculate the relationship between driving force, flux linkage, current and displacement.

As shown in Fig. 7(a), the curved surface of the flux linkage is approximately a plane. This shows that the flux linkage of EMFA has a linear relationship with current or displacement. This result is consistent with the Eqs (4) and (5). As shown in Fig. 7(b), the force surface of the EMFA has a larger rate of change at both ends of the displacement, this is caused by the holding force. In the middle area of the displacement, the curved surface is also approximately a flat surface, this is caused by the ampere force generated by the coil.

Fig. 7.

Curved surface of driving force, flux linkage, current and displacement. (a) Curved surface of flux linkage, current and displacement. (b) Curved surface of driving force, current and displacement.

The above surface is calculated by interpolation in Simulink, and the voltage balance equation of the driving circuit satisfies the Eq. (9). The Runge–Kutta method is used to iteratively calculate the dynamic characteristics of the contactor. Capacitor is used to drive the EMFA. After the displacement control point, PWM with different duty cycles is used for chopping control.

Figure 8 shows the current simulation curve and experimental waveform of EMFA under the control strategy without displacement sensor. The duty cycle combination is 90%, 50%, and the control point is 0.5 mm. When the PWM frequency changes, the drive current drops rapidly so that the EMFA displacement change speed slows down. And can clearly see the sawtooth current generated by the chopping wave.

Fig. 8.

Calculation and experimental results of current.

4.2. Transient characteristic experiment

The displacement prediction of a contactor with a constant air gap has been introduced in detail in the previous section. Based on the control theory and simulation results, this section uses EMFA-driven contactors to carry out related experiments. It is used to verify the actual application effect of the displacement sensorless control method. Based on the novel displacement prediction method, a segmented PWM method is used to verify the effectiveness of bounce suppression. This article mainly conducts experiments by changing the duty cycle combination and the displacement control point. The experimental equipment is shown in Fig. 9. The experimental equipment includes: power module, chopper module, DSP, contactor and sensor. DSP collects voltage and current signals. The PWM signal is generated according to the logic described in the paper. After the PWM signal enters the chopper module, it drives the contactor to act. The power module includes 5 V, 24 V and 72 V voltage outputs, which are used for IGBT drive and capacitor charging respectively.

Fig. 9.

Experimental equipment.

The displacement control point x is fixed at 0.5 mm, change the duty cycle of the second stage of PWM and conduct experiments to get the waveform shown in Fig. 10.

Fig. 10.

Test waveforms of different duty cycle combinations.

As shown in Fig. 10, by increasing the PWM duty ratio of the second stage, it can be clearly seen that the current rises significantly after the displacement of the control point. In this case, the rate of rise of the actuator displacement curve increases, and the bounce time increases. Experiments show that reducing the second stage PWM duty cycle slows down the actuator and reduces the bounce time. The specific experimental data is shown in Table 1. Among them, 𝛼_p is the duty cycle of the first stage, 𝛼_r is the duty cycle of the second stage, x is the displacement control point, t _b is the bounce time, and t _c is the closing time. The data in the table is the average value obtained after 20 tests. Experiments show that adjusting the duty cycle combination can effectively reduce the bounce time of the contacts.

Table 1

Test results of different duty cycle combinations

𝛼p	𝛼r	x/mm	tb/ms	tc/ms
90	50	0.5	0.22	2.84
90	60	0.5	0.90	2.80
90	70	0.5	0.94	2.80
90	80	0.5	1.09	2.71
90	90	0.5	1.12	2.71

Although the closing time of the contactor is prolonged, there is a balance between high speed and low bounce. The relevant parameters of the controller can be adjusted to meet the needs of actual applications.

As shown in Fig. 11, the duty cycle combination is fixed, the first stage is 90%, the second stage is 50%. And the displacement control point is changed to observe the experimental results. As shown in Fig. 11, by extending the displacement control point, it is found that the duration of the large current increases, the actuator movement speed gradually increases, and the bounce time gradually increases.

Fig. 11.

Test waveforms of different displacement control points.

As shown in Fig. 11, under the influence of PWM chopping, the current of the actuator presents a sawtooth shape. When reaching the displacement control point, due to the decrease of the duty cycle, the current drops rapidly and then stabilizes in a lower range. It can be seen from the displacement curve that as the control point moves backward, the displacement jitter gradually increases after the actuator is closed. This is because as the displacement control point moves backward, the energy of the actuator increases during the closing process. As shown in Table 2, the back movement of the displacement control point reduces the closing time and also reduces the bounce time of the contact.

Table 2

Test results of different displacement control points

𝛼p	𝛼r	x/mm	tb/ms	tc/ms
90	50	0.5	0.22	2.84
90	50	1	0.23	2.77
90	50	1.5	1.03	2.72
90	50	2	1.11	2.72
90	50	2.5	1.19	2.70

The same is that the displacement control point can continue to optimize to achieve a balance between closing time and bounce time. If a higher speed is required, the control point should be moved back. And if a smaller bounce time is required, the control point should be forwarded.

As shown in Table 2, the PWM chopping control reduces the contact bounce and closing speed. But when the displacement control point is selected as 1 mm, the bounce time is shorter and has a faster closing time, so the control strategy has a balance point between high speed and low bounce. Simulations and experiments prove that the new displacement sensorless control method can effectively restrain the contact bounce.

5. Conclusion

This paper proposes a new type of dynamic displacement prediction method, it can predict the displacement of the actuator by observing the flux linkage generated by the permanent magnet. And the control method can improve the dynamic characteristics of the contactor with constant air gap. The equivalent magnetic circuit model of EMFA is used to explain the basic principle of displacement prediction. The transient simulation of EMFA is calculated by interpolation method. Under the guidance of simulation, related experiments are carried out for EMFA. Displacement sensorless control reduces the bounce time by 80%. The experiment proves the accuracy of the simulation and the effectiveness of the new dynamic displacement prediction method.

Footnotes

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grants 51877026, in part by State Key Laboratory of Reliability and Intelligence of Electrical Equipment (No. 2020009), Hebei University of Technology.

Nomenclature

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