APPLICATION NOTE R01AN3786EJ0102 Motor Control Application Rev.1.02 Oct. 31,2018 Sensorless Vector Control for Permanent Magnet Synchronous Motor (Algorithm) Summary This application note explains the speed control algorithm in the sensorless vector control software for permanent magnet synchronous motor (PMSM) using Renesas Electronics Corporation’s microcontroller. Contents 1. Overview .......................................................................................................................................... 2 2. PMSM Fundamental Model ............................................................................................................. 2 3. Control System Design ................................................................................................................... 5 4. Sensorless Vector Control ........................................................................................................... 10 R01AN3786EJ0102 Rev.1.02 Page 1 of 21 Oct. 31,2018 Motor Control Application Sensorless Vector Control for Permanent Magnet Synchronous Motor (Algorithm) 1. Overview This application note explains the speed control algorithm in the sensorless vector control software for permanent magnetic synchronous motor (PMSM) using Renesas Electronics Corporation’s microcontroller. 2. PMSM Fundamental Equation 2.1 PMSM Model in Three-Phase (U, V, W) Coordinate Voltage equation of the permanent magnet synchronous motor having sinusoidal magnetic flux distribution (Figure 2-1) can be expressed as follows. U axis iU Ra LU MWU θ MUV N S LV iW Ra LW Ra MVW V axis W axis iV Figure 2-1 Conceptual Diagram of Three-Phase Permanent Magnet Synchronous Motor 푣푢 푖푢 휙푢 [푣푣 ] = 푅푎 [푖푣 ] + 푝 [휙푣 ] 푣푤 푖푤 휙푤 휙푢 퐿푢 푀푢푣 푀푤푢 푖푢 푐표푠휃 [휙푣 ] = [푀푢푣 퐿푣 푀푣푤] [푖푣 ] + 휓 [cos⁡(휃 − 2휋/3)] 휙푤 푀푤푢 푀푣푤 퐿푤 푖푤 cos⁡(휃 + 2휋/3) 푣푢, 푣푣, 푣푤：Stator phase voltage 퐿푢, 퐿푣, 퐿푤：Stator phase self-inductance 푖푢, 푖푣, 푖푤：Stator phase current 푀푢푣, 푀푣푤, 푀푤푢：Mutual inductance 휙푢, 휙푣, 휙푤：Stator phase interlinkage flux 휓：Maximum flux linkage due to permanent magnet 푅푎：Stator phase resistance 휃：Rotor electrical angle from phase U 푝：Differential operator R01AN3786EJ0102 Rev.1.02 Page 2 of 21 Oct. 31,2018 Motor Control Application Sensorless Vector Control for Permanent Magnet Synchronous Motor (Algorithm) 2.2 PMSM Model in Direct-Quadrature (d, q) Coordinate Vector control is a method to control the motor on the two-phase (d, q) coordinate system instead of the three-phase (u,v,w) coordinate system. The d-axis is set in the direction of the magnetic flux (N pole) of the permanent magnet and the q-axis is set in the direction which progresses by 90 degrees (electrical) in the forward direction of the angle θ from the d-axis. d axis Ra id Ld Lq Ra N q axis S iq Figure 2-2 Conceptual Diagram of the Two-Phase Direct Current Motor The coordinate transformation is performed by the following transformation matrix. 2 푐표푠휃 cos⁡(휃 − 2휋/3) cos⁡(휃 + 2휋/3) 퐶 = √ [ ] 3 −푠푖푛휃 −sin⁡(휃 − 2휋/3) −sin⁡(휃 + 2휋/3) 푣푢 푣푑 [ ] = 퐶 [푣푣 ] 푣푞 푣푤 The voltage equation in the two-phase (d, q) coordinate system is obtained as follows. 푣푑 푅푎 + 푝퐿푑 −휔퐿푞 푖 0 [ ] = [ ] [ 푑] + [ ] 푣푞 휔퐿푑 푅푎 + 푝퐿푞 푖푞 휔휓푎 푣푑, 푣푞：d-axis and q-axis voltage 퐿푑, 퐿푞：d-axis and q-axis inductance 푖 , 푖 ：d-axis and q-axis current 3(퐿 − 퐿 ) 3(퐿 + 퐿 ) 푑 푞 퐿 = 푙 + 푎 푎푠 , 퐿 = 푙 + 푎 푎푠 푑 푎 2 푞 푎 2 푅푎：Stator phase resistance 휓a：Flux linkage due to permanent magnet ω：Angular speed 3 휓 = √ 휓 a 2 R01AN3786EJ0102 Rev.1.02 Page 3 of 21 Oct. 31,2018 Motor Control Application Sensorless Vector Control for Permanent Magnet Synchronous Motor (Algorithm) Based on this, it can be considered that alternate current flowing in the stationary three-phase stator is equivalent to direct current flowing in the two-phase stator rotating synchronously with the permanent magnet operating as a rotor. The generated torque can be written as follows from the exterior product of the electric current vector and armature inter- linkage magnetic flux. The first term on the right side of this formula is called magnet torque and the second term on the right side of this formula is called reluctance torque. 푇 = 푃푛{휓푎푖푞 + (퐿푑 − 퐿푞)푖푑푖푞} 푇：Motor torque 푃푛：Number of pole pairs The PMSM which has no difference between the d-axis and q-axis inductances is defined as non-salient PMSM. In this case, as the reluctance torque is 0, the total torque is proportional to the q-axis current. Due to this, the q-axis current is called torque current. In two-phase (d, q) phase coordinate, the d-axis flux is sum of permanent magnet flux and flux generated by d-axis current. Since the equivalent rotating stator flux (in three-phase (u, v, w) coordinate system) is controlled by d-axis current, the d-axis current is called as excitation current R01AN3786EJ0102 Rev.1.02 Page 4 of 21 Oct. 31,2018 Motor Control Application Sensorless Vector Control for Permanent Magnet Synchronous Motor (Algorithm) 3. Control System Design 3.1 Vector Control System and the Controller Speed control block diagram of the vector control is shown below. + v * v i *=0 d dq u d Current v ω* - v PWM M Speed iq* PI + v * v q UVW w PI + vq** vd** θ Sensor ω i i q d Decoupling Control ω θ i i i d d dq u i i iw q q UVW Speed detection Figure 3-1 Vector Control System Block (Speed Control) As shown in Figure 3-1, this system consists of the speed control system and the current control system. These systems use general PI controller. PI controller gains of each system must be designed properly to realize required control characteristics. ∗∗ ∗∗ In decoupling control block, 푣푑 ,⁡푣푞 (as the following equations) are calculated and then added to voltage command value. This realizes the high response of speed control system and enables to control the d-axis and q-axis independently. ∗∗ 푣푑 = −휔퐿푞푖푞 ∗∗ 푣푞 = 휔(퐿푑푖푑 + 휓푎) R01AN3786EJ0102 Rev.1.02 Page 5 of 21 Oct. 31,2018 Motor Control Application Sensorless Vector Control for Permanent Magnet Synchronous Motor (Algorithm) 3.2 Current Control System 3.2.1 Design of Current Control System The current control system is modeled by using the electrical characteristics of the motor. The stator coil can be represented by a resistance R and an inductance L. The stator model of the motor is expressed by the transfer function 1 of the typical RL series circuit . 푅+퐿푠 The current control system model can be represented by a feedback control system using PI control. (Figure 3-2) v i Output Input ∆i Ki 1 Current Current Kp + s R + Ls Y(s) X(s) Controller Stator Model 푅: Resistance of stator coils [Ω] 퐿: Inductance of stator coils [H] 퐾푝: Proportional gain of the current PI control 퐾: Integral gain of the current PI control Figure 3-2 Current Control System Model Based on this model, PI gains of the current control system are designed as the following method. First, the closed-loop transfer function of this system is obtained as follows. 퐾 푠 푎 (1 + ) 푌(푠) 퐾 푎 퐺(푠) = = 푏 ( ) 1 퐾 퐾 푋 푠 푠2 + (1 + 푎) 푠 + 푎 퐾푏 푎 퐾푏 퐾푝푎 퐿 퐾 = 퐾 푎, ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡퐾 = , ⁡⁡⁡⁡⁡⁡⁡⁡퐾 = 푝 푎 푅 푏 푅 The general equation of second-order lag system with zero point can be expressed as follows. 2 휔푛 푠 2 2 (1 + ) 푠 + 2휁ω푛푠 + 휔푛 휔푧 R01AN3786EJ0102 Rev.1.02 Page 6 of 21 Oct. 31,2018 Motor Control Application Sensorless Vector Control for Permanent Magnet Synchronous Motor (Algorithm) By comparing coefficients of two equations above, the following equations are obtained. 푠 퐾 푠 휔2 (1 + ) 푎 ∗ (1 + ) 푛 휔 퐾 푎 푧 ⇔ 푏 2 2 1 퐾 퐾 푠 + 2휁ω푛푠 + 휔푛 푠2 + (1 + 푎)푠 + 푎 퐾푏 푎 퐾푏 2 퐾푎 1 퐾푎 휔푛 = , 2휁ω푛 = (1 + ) , 휔푧 = 푎 퐾푏 퐾푏 푎 From above equations, natural frequency ω푛, damping ratio 휁, zero-point frequency ω푧 are written as follows. 2 퐾푎 1 퐾푎 휔푛퐿 ω푛 = √ , 휁 = (1 + ), 휔푧 = 푎 = 퐾푏 퐾푎 푎 2휁ω푛퐿 − 푅 2퐾푏√ 퐾푏 Current PI control gains (퐾푝_푐푢푟푟푒푛푡,퐾_푐푢푟푟푒푛푡) are written as the following equations. 2 퐾푝_푐푢푟푟푒푛푡 = 2ζ퐶퐺ω퐶퐺퐿 − 푅, 퐾_푐푢푟푟푒푛푡 = 퐾푝_푐푢푟푟푒푛푡푎 = 휔퐶퐺퐿 ω퐶퐺: Desired natural frequency of current control system ζ퐶퐺: Desired damping ratio of current control system Therefore, PI control gains of the current control system can be designed by ω퐶퐺 and 휁퐶퐺. R01AN3786EJ0102 Rev.1.02 Page 7 of 21 Oct. 31,2018 Motor Control Application Sensorless Vector Control for Permanent Magnet Synchronous Motor (Algorithm) 3.3 Speed Control System 3.3.1 Design of Speed Control System The speed control system is modeled by using the mechanical characteristics of the motor. The mechanical system torque equation is written as follows. 푇 = 퐽휔̇ 푚푒푐ℎ 퐽: Inertia of rotor, 휔푚푒푐ℎ: Speed (Mechanical) In consideration of only magnet torque, the electrical system torque equation is written as follows. 푇 = 푃푛휓푎푖푞 By using the mechanical and electrical torque equation, the speed (mechanical) is written as follows. 푃 휓 휔 ＝ 푛 푎 푖 푚푒푐ℎ 푠퐽 푞 The speed in the control software is treated as the electrical speed. Thereby, the number of pole pairs 푃푛 is multiplied to both sides of this equation. 푃 2휓 휔 ＝ 푛 푎 푖 푒푙푒푐 푠퐽 푞 휔푒푙푒푐: Speed (Electrical) The speed control system model can be represented by a feedback control system using PI control. (Figure 3-3) ∆ω iq 2 ω Output Input Ki Pnψa Speed Speed Kp + s Js Y(s) X(s) Controller Plant model Figure 3-3 Speed Control System Model R01AN3786EJ0102 Rev.1.02 Page 8 of 21 Oct. 31,2018 Motor Control Application Sensorless Vector Control for Permanent Magnet Synchronous Motor (Algorithm) Based on this model, PI gains of the speed control system are designed as the following method.