The excellent characteristics of permanent magnet synchronous motors (PMSMs) have led to their widespread application. Therefore, a basic analysis of their working principle is necessary. While there are already analyses and derivations of electromechanical energy conversion for various other motors, there is currently no article directly analyzing the electromechanical energy conversion of PMSMs. This paper uses Maxwell's equations, the law of conservation of energy, and Ampere's law to derive the electromechanical energy conversion of PMSMs. Compared to the energy conversion derivations for other motors, the derivation in this paper is clear and easy to understand.
Keywords: Permanent magnet synchronous motor, electromechanical energy conversion, excitation torque, reluctance torque
1. Introduction
Currently, permanent magnet synchronous motors are being used more and more widely, such as in CNC machine tools, electric bicycles, and electric vehicles. This is due to their unique advantages, including simple control, high power density, and low losses. To better learn, control, and maintain motors, we should understand their working principles. It is known that the mechanical energy of a motor comes from the electrical energy connected to the stator windings; that is, the energy conversion from electrical energy to mechanical energy. This can be derived using Maxwell's equations and the law of conservation of energy. However, current textbooks generally only derive this process for DC motors, electrically excited synchronous motors, and three-phase AC asynchronous motors, without deriving the energy conversion process of permanent magnet synchronous motors. Moreover, the derivation processes for these motors are very complex and difficult to understand. This article will refer to the energy conversion processes of other motors and derive the energy conversion process of permanent magnet synchronous motors in a simple and detailed manner, providing a better understanding when analyzing and applying permanent magnet synchronous motors.
2. Energy conversion of surface-mounted permanent magnet synchronous motors
There are three basic types of permanent magnet synchronous motors: surface-mounted, embedded, and plug-in. Based on their energy conversion mechanisms, they are further divided into two categories: surface-mounted motors are a separate category, while embedded and plug-in motors are combined into one. For simplicity, we will use a cross-sectional view of a single-phase surface-mounted permanent magnet synchronous motor (Figure 2-1) as an example.
Figure 2-1
In this electromechanical system, due to the uniform air gap, the self-inductance of the stator winding is constant. Taking the entire motor as a system, assuming that the initial energy consists only of the magnetic field energy generated by the permanent magnet, at t=0, the stator is given a short dt time interval. The motor rotor then rotates at a constant speed through a very small angle, neglecting losses and leakage flux. According to the law of conservation of energy, for the entire motor system, the input electrical energy + the magnetic field energy generated by the permanent magnet = the internal energy generated by the stator winding + the residual magnetic energy of the stator winding + the magnetic field energy generated by the permanent magnet + the rotor mechanical energy.
Both sides of the equation contain magnetic field energy generated by the permanent magnets, which is the same magnetic field energy throughout the entire motor system. Therefore, the formula becomes: Input electrical energy = Internal energy generated by the stator windings + Residual magnetic energy of the stator windings + Rotor mechanical energy. This can be expressed mathematically as follows:
(2-1)
It is the stator input electrical energy, the internal energy of the stator windings, the magnetic energy of the stator windings, and the rotor mechanical energy.
(2-2)
(2-3)
(2-4)
(2-5)
As described above, part of the electrical energy input to the stator winding is consumed by the resistance in the winding, and part is stored in the air gap through the magnetic field energy of the coil and converted into the mechanical energy of the rotor. Therefore,
(2-6)
It is the initial magnetic field energy of the stator inductor (or the total magnetic field energy of the stator).
(2-7)
It is an induced electromotive force; changes in the magnetic flux will inevitably cause changes in the voltage of the coil.
(2-8)
According to equations (2-7) and (2-8), we have
(2-9)
It is the total magnetic flux linkage of the stator winding, that is, the effective magnetic flux linkage that passes perpendicularly through the coil, so we can obtain...
(2-10)
According to equations (2-1) and (2-6), we have
(2-11)
According to equations (2-4) and (2-9), we have
(2-12)
(2-13)
3. Derive the torque generation of a surface-mounted permanent magnet synchronous motor using Ampere's law.
We learned Ampere's law in high school, and the formula for the force on an electric current in a magnetic field is (3-1).
In a permanent magnet synchronous motor system, the current is the current in the stator windings, and the magnetic field comes from the permanent magnets of the rotor. Therefore, the stator windings are subjected to a force, but they cannot move. According to Newton's third law, the rotor will also be subjected to a force of the same magnitude but in the opposite direction. The torque of the rotating body is equal to the force multiplied by the radius of rotation. Therefore, the electromagnetic torque can be calculated.
According to the theory, only one magnetic field exists in the entire motor system, which is the magnetic field generated by the rotor permanent magnet. According to Gauss's law of magnetic fields, when , , , because, therefore we have (this is an assumption). But this only holds true when in use.
Assuming the stator current is v, the number of turns is v, the magnetic field strength perpendicular to the stator winding is v, and there is no leakage flux, the force on one side of the stator winding is v.
(3-2)
The excitation flux density is (3-3).
The reason for the sine wave is that it passes through the conductor rather than the coil, because (3-4) is the stator inner diameter, the pole pitch, and the effective length of one turn of the stator winding on one side. According to equations (3-3) and (3-4), we have...
(3-5)
Substituting equation (3-5) into equation (3-2), we get
(3-6)
(3-7)
4. Motor energy conversion of embedded and plug-in permanent magnet synchronous motors
Figure 4-1
The structure of the embedded permanent magnet synchronous motor is shown in Figure 4-1. The permanent magnet is embedded in the surface of the rotor core. When the rotor position is as shown in Figure 4-1a, the self-inductance of the stator winding is and the total magnetic flux is (4-1). When the rotor position is as shown in Figure 4-1c, the self-inductance of the stator winding is and the total magnetic flux is (4-2).
Because the air gap magnetic circuit is different, therefore...
When the rotor is in other positions, as shown in Figure 4-1b, the self-inductance of the stator winding is (4-3).
This result was obtained by considering only the fundamental component, but it is sufficient for analysis and engineering applications. The total magnetic flux linkage of the stator is...
(4-4)
Assuming initially there is no electrical energy input, and then electrical energy is input for an extremely short time period dt, the rotor rotates at a constant speed by a very small angle. According to the law of conservation of energy,
(4-5)
(4-6) Because
(4-7)
Therefore (4-8)
(4-9)
The negative sign in equation (4-9) relates to the direction of rotation and can be removed. This torque comprises two terms: the first is the excitation torque, obtained from the force exerted on a current-carrying conductor in a magnetic field; the second is the reluctance torque, caused by the unevenness of the air gap.
5. Derive the generation of embedded and insertion torques using Ampere's law.
The formula for deriving torque generation can only derive the generation of excitation torque. The excitation torque of embedded and inserted types is exactly the same as that of surface-mounted types, and the derivation process is the same. However, the Ampere force formula cannot be used to derive reluctance torque. The Ampere force formula means that a current conductor is subjected to a force when cutting magnetic field lines.
6. Summary
The electromechanical energy conversion of a permanent magnet synchronous motor (PMSM) satisfies Maxwell's equations and the law of conservation of energy. The energy for the motor's rotation originates from the electrical energy at the stator terminals. The stator windings generate magnetic field energy, part of which is converted into mechanical energy, and part remains as air gap magnetic field energy. This is true for all types of PMSMs. However, the reluctance torque in the torque of particularly embedded and plug-in PMSMs cannot be derived from this, because it is not the torque caused by the force exerted on the conductor's current in the magnetic field.
References:
Wang Chengyuan, Xia Jiakuan, Sun Yibiao. Modern Motor Control Technology [M]. Beijing: China Machine Press, 2009, pp. 1-13.
Nannapaneni Narayana Rao. Fundamentals of Engineering Electromagnetism [M]. Translated by Zhou Jianhua and You Baiqiang. Beijing: China Machine Press, 2006, 48-116; 209-216.
