Special topic of rotor position estimation of permanent magnet synchronous motor -- Summary of position estimation of fundamental wave model

introduction

This paper classifies the methods of estimating the rotor position through the fundamental wave model of permanent magnet synchronous motor , Summarize the angle estimated by back EMF 、 Estimate the angle through the rotor flux 、 Closed loop scheme , Three different ideas , Break it down one by one , Its core principles are described respectively , It also outlines the different ideas of different schemes , Give relevant papers for reference .

1、 Back EMF method

The last article deduced $\alpha \beta$ Voltage equation in coordinate system
$$\left[\begin{array}{c}{v}_{\alpha }\\ v_{\beta }\end{array}\right]=\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]R_s+\frac{d}{dt}\left[\begin{array}{c}{L}_s{i}_{\alpha }\\ L_s{i}_{\beta }\end{array}\right]+\left[\begin{array}{c}-{\omega }_e{\varphi }_{f}sin\theta \\ {\omega }_e{\varphi }_{f}cos\theta \end{array}\right]\text{\hspace{1em}(1)}$$
among ${\omega }_e$ Is the electrical angular velocity .
Easy to see ,$\alpha \beta$ The voltage consists of three parts , Resistance voltage drop , The voltage produced by the change of inductive current , The back electromotive force produced by the change of the flux linkage of the equivalent permanent magnet . As mentioned above , The rotor position of permanent magnet synchronous motor can be extracted from the back EMF . The key point of various position estimation methods of back EMF is to separate the back EMF term from the voltage equation . We are running senseless foc when , Motor stator resistance , Motor stator inductance , The flux linkage constant of the permanent magnet is a known constant , Voltage and current are measurable parameters , speed 、 The rotor position is the parameter to be estimated . For the three components of the voltage equation , The resistance voltage drop part is easy to calculate , The inductance voltage cannot be accurately calculated because it involves differentiation , Although the voltage is known , However, the direct calculation error of inductance voltage is large , It is difficult to directly separate the components of back EMF .
For this kind of content that is difficult to calculate directly , It is often estimated by designing observers .
$$\frac{d}{dt}\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]=\left[\begin{array}{c}{v}_{\alpha }\\ v_{\beta }\end{array}\right]\frac{1}{L_s}-\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]\frac{R_s}{L_s}+\left[\begin{array}{c}-{\omega }_e{\varphi }_{f}sin\theta \\ {\omega }_e{\varphi }_{f}cos\theta \end{array}\right]\frac{1}{L_s}\text{\hspace{1em}(2)}$$
The written voltage equation is in the above form , For the rightmost back EMF term , Not easy to calculate , As an unknown part , Current equals formula 2 The integral on the right . There may be an error between the current obtained by integration and the real current sampled , use $er{r}_{\alpha },er{r}_{\beta }$ Express .
$$\left[\begin{array}{c}{\hat{i}}_{\alpha }\\ {\hat{i}}_{\beta }\end{array}\right]=\left[\begin{array}{c}{v}_{\alpha }\\ v_{\beta }\end{array}\right]\frac{1}{L_s}-\left[\begin{array}{c}{\hat{i}}_{\alpha }\\ {\hat{i}}_{\beta }\end{array}\right]\frac{R_s}{L_s}+\left[\begin{array}{c}{x}_{\alpha }+er{r}_{\alpha }\\ x_{\beta }+er{r}_{\beta }\end{array}\right]\frac{1}{L_s}\text{\hspace{1em}(3)}$$
among
$er{r}_{\alpha }=i_{\alpha }-{\hat{i}}_{\alpha };er{r}_{\beta }=i_{\beta }-{\hat{i}}_{\beta }$
Use some form of correction , Such as synovium , send $er{r}_{\alpha }=0,er{r}_{\beta }=0$, be
$$\left[\begin{array}{c}{x}_{\alpha }\\ x_{\beta }\end{array}\right]=\left[\begin{array}{c}-{\omega }_e{\varphi }_{f}sin\theta \\ {\omega }_e{\varphi }_{f}cos\theta \end{array}\right]$$
Here we are , Complete the separation of the back EMF term , adopt arctan Or phase locked loop calculation can obtain the rotor position , For speed , It can be obtained by phase-locked loop , It can also be obtained by positional differentiation , in addition ,$$\begin{gathered} \sqrt{x_{\alpha }^2+ \\ {x}_{\beta }^2}/{\varphi }_f \end{gathered}$$ You can also get the rotor speed .

Related articles ：
Kim H, Son J, Lee J. A high-speed sliding-mode observer for the sensorless speed control of a PMSM[J]. IEEE Transactions on Industrial Electronics, 2010, 58(9): 4069-4077.
observation $i_{\alpha },i_{\beta }$, Use the traditional synovial observer Signum Function Do the switch function , This article USES the sigmoid function Do the switch function , Low pass filtering can be avoided .
Qiao Z, Shi T, Wang Y, et al. New sliding-mode observer for position sensorless control of permanent-magnet synchronous motor[J]. IEEE Transactions on Industrial electronics, 2012, 60(2): 710-719.
This article is optimized on the basis of the previous article , A speed observer is designed , Instead of calculating speed by differentiating angles .

2、 Flux linkage method

The last article mentioned $\alpha \beta$ The following expression of the voltage equation in the coordinate system .
$$\left[\begin{array}{c}{v}_{\alpha }\\ v_{\beta }\end{array}\right]=\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]R_s+\frac{d}{dt}\left[\begin{array}{c}{\varphi }_{\alpha }\\ {\varphi }_{\beta }\end{array}\right]\text{\hspace{1em}(4)}$$
among ${\varphi }_{\alpha }{\varphi }_{\beta }$ Express $\alpha \beta$ Axial flux linkage
$$\left[\begin{array}{c}{\varphi }_{\alpha }\\ {\varphi }_{\beta }\end{array}\right]=\int \left(\left[\begin{array}{c}{v}_{\alpha }\\ v_{\beta }\end{array}\right]-\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]R_s\right)dt\text{\hspace{1em}(5)}$$
Finishing type 4, Get the formula 5, The flux linkage can be written in addition to the resistance voltage drop , Integration of other voltages . Imagine a coil , Placed in a magnetic field , When the magnetic field changes , Voltage will be generated at both ends of the first coil . Easy to understand formula 4, Understand the formula 4, It is easy to deduce the formula 5, type 5 Calculate the flux linkage through the integration of voltage , This formula can be called the voltage model of permanent magnet synchronous motor flux linkage model ..

Above $\alpha \beta$ The flux linkage is the total flux linkage on two axes , The flux linkage of each shaft can be written as the combination of rotor part flux linkage and stator part flux linkage . stay $dq$ Coordinate system ,$d$ Shaft is defined as rotor permanent magnet N The direction of the pole , So the rotor permanent magnet flux linkage ${\varphi }_f$ All in $d$ Axis , meanwhile $dq$ The shaft also has stator flux .
As mentioned earlier, the change of magnetic field will generate voltage at both ends of the coil , The stator inductance of permanent magnet synchronous motor is such a coil .

from $\varphi =\int vdt$; And $L\frac{di}{dt}=v$; Available $\varphi =Li$; That is, the magnetic chain generated by the current flowing through the coil is the product of inductance and current , if $dq$ The shaft current is not 0,$dq$ The shaft flux linkage will contain stator flux linkage components , adopt ipark transformation $\alpha \beta$ Axial flux linkage , As follows .
$$\left[\begin{array}{c}{\varphi }_{\alpha }\\ {\varphi }_{\beta }\end{array}\right]=\left[\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right]\left[\begin{array}{c}{L}_s{i}_d+{\varphi }_f\\ L_s{i}_q\end{array}\right]\text{\hspace{1em}(6)}$$
This formula is calculated by the current flowing through the inductance $\alpha \beta$ Axial flux linkage , It can be called the current model of permanent magnet synchronous motor flux model .

Because the rotor position information is contained in the rotor flux linkage component , And this part is easy to pass $\alpha \beta$ Separation of flux linkage equations in coordinate system , The position estimation methods of flux linkage are mostly $\alpha \beta$ Calculate the total flux linkage in the coordinate system , Then subtract the stator flux , Get the rotor flux , Then the rotor position and speed are obtained by using phase-locked loop , Or obtain the rotor position by arctangent , Then the differential gives the velocity .

It is worth mentioning that the rotor flux linkage has nothing to do with the motor speed , Using the flux linkage method to estimate the rotor position of the motor is easy to obtain better low-speed performance .

To accurately separate the rotor flux components is a core problem of the flux method , The common way is to calculate the total flux linkage through the voltage and current model , Then take a reference value that is not easy to change to correct the error . Or the famous reference model adaptive method , Take the current model as a reference , Voltage model adaptation ; The two models can also be fused directly , Trust the current model at low speed , Trust the voltage model at high speed .

The way of magnetic linkage is 2010 The more popular method after years , There are many points worth discussing in detail , Relevant content will be analyzed in a subsequent meeting .

Related articles ：
Lee J, Hong J, Nam K, et al. Sensorless control of surface-mount permanent-magnet synchronous motors based on a nonlinear observer[J]. IEEE Transactions on power electronics, 2009, 25(2): 290-297.
Famous nonlinear observer , Use the rotor flux to correct the voltage model .

An L, Franck D, Hameyer K. Sensorless control for surface mounted permanent magnet synchronous machines at low speed[C]//2013 International Conference on Electrical Machines and Systems (ICEMS). IEEE, 2013: 77-82.
Use current model to correct voltage model , Or use the sampling current to correct the voltage model .

3、 Closed loop method

First look at $dq$ Coordinate system voltage equation
$$\begin{gathered} u_d=R_s{i}_d+L_d\frac{d{i}_d}{dt}-{\omega }_e{L}_q{i}_q \\ {u}_q=R_s{i}_q+L_q\frac{d{i}_q}{dt}+{\omega }_e{L}_d{i}_d+{\omega }_e{\varphi }_f\text{\hspace{1em}(7)} \end{gathered}$$

$dq$ The idea of position estimation in coordinate system is essentially different from that mentioned above , The two methods mentioned above belong to open-loop schemes , The input of the position observer is $i_{\alpha },i_{\beta };u_{\alpha },u_{\beta }$, The output of the observer is angle and speed . The output of the observer will not affect the input in turn .

however $dq$ The scheme under the coordinate system is different , This kind of scheme requires that the input is calculated by using the angle of the observer output $i_d,i_q;u_d,u_q$, And then use $i_d,i_q$ Calculated by voltage equation ${\hat{u}}_d,{\hat{u}}_q$, Or use $u_d,u_q$ Calculated by voltage equation ${\hat{i}}_d,{\hat{i}}_q$, The estimated value is compared with the sampled value , If the angle is more accurate , The error is relatively small , Or on the contrary, if the estimation error tends to 0, Then the output angle of the observer converges near the real angle .

To look back $dq$ Voltage equation in coordinate system , Back EMF component at rated speed ${\omega }_e{\varphi }_f$ Generally, it can occupy $q$ Shaft voltage 70% above , If the angle is accurate , The back EMF components will all lead out at $q$ Axis , If there is an error in the angle ,$d$ The shaft will have a back EMF component , Calculated using the voltage equation $d$ The shaft voltage will be significantly less than park Transformed $d$ Shaft voltage . Closed loop methods often take similar errors as inputs , Design regulator , The output of the regulator is speed or angle , Adjustment makes the error approach 0, The angle is the exact angle .

There are also many places worth discussing in this kind of scheme , The following articles will be analyzed in detail .

Related articles ：
Lascu C, Andreescu G D. PLL position and speed observer with integrated current observer for sensorless PMSM drives[J]. IEEE Transactions on Industrial Electronics, 2020, 67(7): 5990-5999.
2020 New paper in , Use a current observer to estimate $dq$ electric current , The angle and speed errors are reflected by current errors .

Harnefors L, Ottersten R. Regenerating-Mode Stabilization of the “Statically Compensated Voltage Model”[J]. IEEE Transactions on Industrial Electronics, 2007, 54(2): 818-824.
Good low-speed performance is achieved

4、 Summary

It says 3 The specific principle of class method , According to my understanding and observation of the industry , Back EMF method was the mainstream method in the early years , So far, the scheme of high-speed motor control is also used , This kind of method is difficult to achieve good low-speed performance . The method of magnetic linkage is the mainstream in recent years , Many common platforms such as ti fast The scheme uses the magnetic linkage method , This kind of method is easy to achieve better low-speed performance . The closed-loop method is a new scheme , It can also achieve better low-speed performance , Online transmission Huichuan has turned to this kind of scheme , According to my judgment of industry trends , This kind of method will be applied more in the future .