Vector Control Simulation of Permanent Magnet Synchronous Linear Motor

Compared with rotating motor, linear motor can “directly” obtain linear motion, which eliminates the intermediate transformation link and can directly drive the equipment requiring linear motion. It is applicable to rotating motor mature control strategy can be better for linear motor control research. Its development and application of linear motor has great significance. In this thesis, starting from the weakening characteristics of permanent magnet synchronous linear motor, through the MATLAB simulation and comparative analysis of different control signals (SPWM and SVPWM) of vector control strategy, the conclusion that vector control strategy can effectively control ideal linear motor is obtained, which lays a foundation for further improving the control strategy to control non-ideal linear motor.


Introduction
With the development of the economy, industrial equipment updated day by day, some special areas of the traditional rotating motor plus mechanical transmission elements to obtain linear motion of the practice is increasingly unable to meet the needs of modern industrial production.The linear motor can "directly" obtain linear motion, eliminating the intermediate transformation link, to provide linear motion to the controlled object in the form of thrust, in order to obtain a unidirectional or bi-directional limited controllable displacement .At present, the research of linear motor is still in the stage of qualitative analysis of characteristics and mathematical model analysis, while the control of linear motor application research is relatively small [1][2][3].Whether the existing mature control strategies can be applied to the control of linear motor needs to be verified by simulation and experiment.The purpose of this thesis is to analyze the mathematical model of permanent magnet synchronous linear motor and control the permanent magnet synchronous linear motor with weakened characteristics by using a mature control strategy (vector control).Comparing the consistency of the simulation results with the theoretical analysis, the feasibility of the current control strategy for the ideal permanent magnet synchronous linear motor is judged, in order to prepare for the study of the current control strategy for the control of the non-ideal permanent magnet synchronous linear motor.

Mathematical Modelling
In the control of motors, it is usually assumed that the magnetic field is sinusoidal distributed, the three-phase windings are symmetrical and the electromagnetic thrust of the motor is proportional to the q-axis current [4].The three-phase windings are not symmetrical due to end effects.Therefore, the mathematical model of the permanent magnet synchronous linear motor needs to be re-established.Figure 1 represents the structural model of a single-sided, flat-plate type permanent magnet synchronous linear motor.As shown in figure 1, a, b, c denote the axes of the three-phase windings, d and q denote the axes of the permanent magnet field, as well as x is the position of the actuator.Literature derived the magnetic chain equation, voltage equation and electromagnetic thrust equation of the linear motor by taking into account the end effects and magnetic field harmonics of the linear motor in establishing the mathematical model of the permanent magnet synchronous linear motor.
ψ and q ψ are the current and the magnetic chain of the d and q axes of the linear motor, respectively.f i is the equivalent current of the permanent magnet.A(x), B(x), C(x) and D(x) are the self-inductance coefficients and mutual inductance coefficients of the d-axis and the q-axis as a function of the position x of the mover, which contains even harmonic components, when the three-phase windings of the motor are asymmetrical [5].E(x) and F(x) are the mutual inductance coefficients between the shaft and the equivalent excitation winding of the permanent magnet, which contain odd harmonic components.There is coupling between the d-axis and q-axis inductances.d u and q u are the d and q-axis voltage.R is the internal resistance of the winding.For τ π ω , e ω is the electrical angular velocity corresponding to the linear motor, v is the linear velocity of the linear motor, and τ is the pole pitch of the linear motor.F is the electromagnetic thrust force on the actuator.

Simulation Model
The mathematical model of the permanent magnet synchronous linear motor is described above, and the following assumptions are made to simplify the simulation.
(1) The three-phase windings of the motor are completely symmetrical; (2) The end effect magnetism is not considered, and various losses are not taken into account; (3) The magnetic field in the air gap is distributed sinusoidal in space.
According to the assumption that A(x)=D(x)=L and B(x)=C(x)=0 for linear motor, the d-q axis model of the permanent magnet synchronous linear motor can be developed [6].
The direct axis voltage equation is: The electromagnetic thrust equation is: Where L is the armature shaft inductance, p is the pole pair number, R is the armature resistance, f ψ is the stator magnetic chain coupled in the armature, Kt= f K ψ ⋅ is the thrust coefficient, v is the motor speed, and Fm is the electromagnetic thrust.
According to equations ( 4) and ( 5), the simulation modules for linear motor with straight and cross axis voltage equations are built separately, as shown in figure 2 and figure 3. Furthermore, the simulation module of permanent magnet synchronous linear motor is established, as shown in figure 4.

Simulation Analysis
Sinusoidal pulse width modulation (SPWM) is used to control motor systems with simple circuit structure and low cost.However, the system performance is not high and has poor stability due to inverter dead band problem [7][8].The space vector pulse width modulation (SVPWM) technique simultaneously controls the state of three-phase currents with the aim of approximating the ideal magnetic field trajectory of the motor air gap, which is capable of improving the power factor and reducing losses.In this section, SPWM based and SVPWM based linear motor control models are developed and analyzed using system simulation software MATLAB respectively.

Simulation Analysis of Sinusoidal Pulse Width Modulation
In order to achieve the precise positioning control of the permanent magnet synchronous linear motor, this model uses vector control (based on SPWM) of the permanent magnet synchronous motor for d i and q i respectively.The SPWM control block diagram is shown in figure 5.As can be seen from the above figure, the vector control has both position and current closed loop.The feedback quantity of current includes two components d i and q i .The two feedback deviations d i and q i achieve the control of the motor.The scale provides the position signal P of the actuator, which is converted to θ and is involved in the current closed loop.At the same time, the position signal P is directly fed back to the position port to realize the closed-loop control of the position.By controlling the voltage component d u , which is in phase with the actuator chain, so that d i ≡ 0.
At this time, the electromagnetic thrust obtains the maximum value.The electromagnetic thrust is controlled by adjusting the cross-axis current q i , thus realizing the decoupling of the control parameters of the permanent magnet synchronous linear motor and achieving the control of the electromagnetic thrust.This control method of permanent magnet synchronous linear motor is called magnetic field oriented vector control [9][10].In magnetic field oriented vector control, it is necessary to obtain the position angle θ of the actuator at any time.The relationship between θ and the position P of the actuator is: θ= τ π / P .τ is the pole distance.The current space vector diagram and Simulink simulation SPWM waveform generation module diagram is shown in figure 6   As shown in figure 8, it can be seen that the permanent magnet synchronous linear motor actuator can track the given position (200mm) well and the tracking time is about 0.5s.

Simulation Analysis of Space Vector Pulse Width Modulation
The simulation block diagram of permanent magnet synchronous linear motor based on space vector pulse width modulation position servo control is shown in figure 9.As shown in figure 10, the embedded MATLAB language programming feature of MATLAB is utilized to program the determination of the number of the sector in which an arbitrary spatial voltage vector is located during motor operation.The allocation times T1, T2 of the operating voltage vectors in the judged sectors are calculated; the corresponding off states of the controllable power electronics are determined; and the SVPWM waveforms for each phase are obtained.As shown in figure 11, it can be seen that the permanent magnet synchronous linear motor actuator can track (about 0.1s) the given position faster.

Simulation Comparison
The above two pulse width modulation model simulations are available.period.Since the motor parameters and the inverter voltage amplitude are equal during the simulation, the power utilization efficiency (current amplitude) of the SVPWM control is significantly higher than that of the SPWM control.As shown in figure 12 and figure 13, the chain trajectory graph shows that the SVPWM control is closer to the ideal magnetic field than the SPWM.

Conclusion
The vector control is to simplify the strong coupling of AC motor to achieve the control effect similar to DC motor by decoupling the relationship between the control quantity and the controlled quantity.Current feedback tracking control belongs to incomplete decoupling control though.However, as long as a larger gain is selected, * d d i i = and * q q i i = can be approximated in order to reduce the current error to an acceptable level.Meanwhile, the control method is simple.
The study of the control of permanent magnet synchronous linear motor needs to start with a simpler model.In this thesis, it is demonstrated through simulation that vector control can effectively control an ideal permanent magnet synchronous linear motor, and the simulation effects of different control signals (SPWM, SVPWM) are further compared.This further confirms the feasibility of vector control for linear motor and lays the foundation for further research on the control of permanent magnet synchronous linear motor.
In this thesis, the simulation focuses on the comparison of the results of different control signals (SPWM, SVPWM) with the theoretical compatibility, while neglecting the optimization analysis of the control itself.

Figure 2 .
Figure 2. Simulation block diagram of direct axis voltage equation.

Figure 3 .
Figure 3. Simulation block diagram of cross-axis voltage equation.

Figure 4 .
Figure 4. Simulation block diagram of simulation of permanent magnet synchronous linear motor.

Figure 5 .
Figure 5. Block diagram of vector control of permanent magnet synchronous linear motor.
and figure 7.

Figure 9 .
Figure 9. Block diagram of SVPWM vector control of permanent magnet synchronous linear motor.

Figure 10 .
Figure 10.Simulation of SVPWM waveform implementation.The simulation parameters are the same as sinusoidal pulse width modulation (SPWM) and the simulation results are as follows.

Figure 13 .
Figure 13.Magnetic chain trajectory under SVPWM control.Above are the phase current waveforms and chain trajectories of the permanent magnet synchronous linear motor for both SPWM and SVPWM controls during the effective control time