Vector control of permanent magnet synchronous motor (I) -- mathematical model

Hello everyone , I meet you again , I'm your friend, Quan Jun .

notes ：1： This is one of the articles on permanent magnet synchronous control , At your request , A series of articles on permanent magnet synchronous vector control have been placed at the top of the home page , You can go directly to the home page , I hope I can help you , thank you . 2： After six articles on vector control . Weak magnetic field 、MTPA、 The explanation of position control series has been supplemented , Also on the home page , Please refer to . 3: Just have a meal , I also made a set of more detailed tutorials and put them on the top , Including basic double closed loop 、MTPA、 Weak magnetic field 、 Three closed loops 、 Fuzzy PI And other basic control optimization strategies , Also slip form ,MRAS Wait for no speed control topics to be sorted out , Please check _**

My tutor's research topic is the control of permanent magnet synchronous motor , The first task assigned to me is to build the simulation of its vector control system . This paper records the learning process of vector control system . Because it is a beginner, my understanding may not be enough , The source of each content will be marked in the article , You can refer to the original .

1、 Mathematical model of permanent magnet synchronous motor （ Refer to Xie Xiaogang 、 Chen Jin adopts Id=0 Permanent magnet synchronous motor vector control article ） Permanent magnet synchronous motor is a nonlinear system , With multivariable 、 Characteristics of strong coupling . We have the following assumptions when analyzing it ：

• Ignore core saturation , Eddy current and hysteresis losses are not included
• Ignore armature reaction during commutation
• Undamped winding on rotor , The permanent magnet has no damping effect
• The magnetic field generated by the permanent magnet and the induced magnetic field generated by the three-phase winding are sinusoidal
• The stator winding current only produces sinusoidal magnetic potential in the air gap , No higher harmonics
• Model according to motor application Under this ideal condition ： 1.1 Stator voltage equation of permanent magnet synchronous motor in three-phase static coordinate system ：（ There is a mistake in the figure below , Stator flux linkage requires one conductor ）

In style Rs Is the armature resistance ,ψa ψb ψc Respectively abc Three phase flux linkage ,ia ib ic They are abc Three phase phase current . 1.2 Magnetic linkage equation in three-phase stationary coordinate system

among Laa、Lbb、Lcc Self inductance of each phase winding , And Laa=Lbb=Lcc, In style Mab Etc. are mutual inductance between windings and are equal .ψf It's a permanent magnet flux linkage ,θ For rotor N Extreme harmony a Angle between phase axes . after CLARK and PARK After the left transform , Get it in dq Mathematical model in coordinate system ： 1.3 dq Voltage equation in coordinate system

among ud、uq by dq Shaft voltage ,id、iq by dq Shaft current ,ψd、ψq by dq Axial flux linkage ,Ld、Lq by dq Shaft inductance ,we Is the speed . 1.4 dq Magnetic linkage equation in coordinate system

1.5 Torque equation

From above 1.5 The transfer moment equation shows that , Electromagnetic torque consists of two parts , The first one is the interaction between permanent magnet and stator winding flux , The second term is caused by the change of magnetoresistance . Here we need to distinguish between salient pole and salient pole motors , Hidden pole motor due to Lq=Ld, Therefore, the variable torque of reluctance is unique to salient pole motor , We also need to pay attention to the motor type when building the simulation . Summary ： The mathematical model of permanent magnet synchronous motor explains its internal structure , Help us design control strategies , We do coordinate transformation and PI It is necessary to analyze its mathematical model when adjusting parameters , Very important , Very important , Very important , Say it three times ！！！

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