Sensorless Control of SPM Motor for e-Bike Applications

1. Introduction

The widespread use of Electric Vehicles (EVs) in urban areas is inevitable, as the adoption of zero-emission vehicles is becoming mandatory for well-known reasons. This transition will be gradual due to the necessary infrastructure requirements, but it is already recognized that EVs will be the future solution for urban transportation [1]. Surface-mounted permanent magnet (SPM) synchronous motors are an ideal choice for applications demanding high power efficiency. The absence of excitation currents and the high power factor make them highly competitive [2]. Electric vehicles used in urban transportation require immediate and highly effective control to prevent any undesirable situations [3]. In sensored Field Oriented Control (FOC) the rotor position information is provided by a mechanical sensor, which leads to a higher cost and lower reliability [4]. Accurate rotor position is required for both open-loop and closed-loop control. Shaft-mounted sensors face challenges such as mechanical faults, decreased operational reliability, uneven placement of position sensors, and operational failures [5]. However, the mechanical sensors can be replaced with a sensorless algorithm that provides rotor position and speed. The sensorless motor control algorithm is a compelling research area in the motor drive sector. Although motor position sensors are still widely used in most industrial motor drives, cost considerations are driving the industry to eliminate them. Additionally, mounting position sensors on machines is often problematic, especially in specific applications. For instance, in outer rotor machines with limited space, attaching the position sensor to the outer cup rotor is challenging. In drones, where the rotor blade motor is typically a small outer rotor machine, there is no room for a high-accuracy position sensor or even a simple hall sensor [6]. For medium and high speed regions, the most used sensorless algorithm are back-EMF based, where the rotor position is extracted from the fundamental wave of the back-EMF [7]. In this paper, the mathematical equations of the SPM motor in the rotating reference frame are first described. Then, FOC control system and Space Vector Modulation (SVM) technique with sensors is explained. A current loop controller with a sensorless algorithm is designed for the SPM motor based on the given specifications. The back-EMF is estimated using frequency-locked loop and phase-locked loop analysis with a Second Order Integral Flux Observer (SOIFO) filter. Finally, electromechanical speed ${\omega }_m^e$ and position ${\theta }_m^e$ are estimated using the quadrature signal generalized phase-locked loop method. A brief conclusion is drawn by analyzing the results at different loads and speeds.

2. Standard Control of SPM Motor

2.1. Permanent Magnet Synchronous Motor

The Permanent Magnet Synchronous Motor (PMSM) is described by the following equations in the rotating reference frame,

$$v_d=R_s{i}_d+L_d{\displaystyle \frac{d{i}_d}{dt}}-{\omega }_m^e{L}_q{i}_q$$

(1)

$$v_q=R_s{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+{\omega }_m^e{L}_d{i}_d+{\omega }_m^e{\mathsf{\Lambda }}_m$$

(2)

where $i_d$ and $i_q$ are the phase current, $v_d$ and $v_q$ are the phase voltage, respectively. ${\omega }_m^e$ is the electromechanical speed, ${\mathsf{\Lambda }}_m$ is the flux linkage produced by the magnets, $L_d$ and $L_q$ are the d-axis and q-axis synchronous inductances and $R_s$ is the phase resistance.

2.2. Field Oriented Control

FOC is the most used control for PMSM. For a given current, maximum torque can be achieved when the stator flux due to the stator current is orthogonal to the rotor flux, created by the PM [8]. Electromagnetic torque is a function of the angle between the stator and rotor flux. The equation that describes the average electromagnetic torque is,

$$T_{avg}={\displaystyle \frac{3}{2}}p{\mathsf{\Lambda }}_m{I}_q$$

(3)

where $I_q$ is the quadrature current, p is the number of pole pairs. Figure 1 shows the FOC architecture. The 3-phase currents and the rotor angle are measured on motor terminals and rotor shaft respectively. The 3-phase currents are converted into a two phase vector frame, $I_{\alpha }$ and $I_{\beta }$, in the stationary state, using the Clarke transformation, then to the rotating reference frame using the Park transformation [9]. The currents are referred to as direct axis current, $I_d$, and quadrature axis current, $I_q$. In order to have the maximum torque for a given current, the current $I_q$ must be fully oriented along the quadrature axis [10]. These two currents are compared to the reference values and the errors are used as input to PI regulators. The control is implemented in the steady state frame, since the currents and voltages are constant. The output of the regulators, which are the direct voltage, $v_d$, and quadrature voltage, $v_q$, undergoes the transformation to the stationary reference frame, and the $\alpha$ and $\beta$ voltages are used to calculate the inverter duty cycle using the space vector modulation (SVM).

The FOC consists in controlling the stator currents and the torque. The idea behind the operation is the transformation from a three-phase time dependent system into a two coordinate system [11].

2.3. Space Vector Modulation

SVM is a modulation technique able to generate the desired phase voltages [12]. Since only one mosfet is ON and the other is OFF in each leg, there are 8 possible combinations. The phase vectors divide the plan into 6 sectors, each spanning ${60}^{\circ }$. Depending on the sector addressing the desired phase voltage, two adjacent vectors are applied in succession. The two vectors are time weighted in a switching period $T_s$ to produce the desired output voltage. Let’s assume the desired voltage is in the first sector $(0^{\circ }-{60}^{\circ })$, the following condition holds

$$\overline{v}={\overline{v}}_{100}{\displaystyle \frac{t_1}{T_s}}+{\overline{v}}_{110}{\displaystyle \frac{t_2}{T_s}}+{\overline{v}}_{111}{\displaystyle \frac{t_0}{T_s}}$$

(4)

where $t_1$ and $t_2$ are the times in which the vectors ${\overline{v}}_{100}$ and ${\overline{v}}_{110}$ are applied respectively, $t_0$ is the time in which the zero vector is applied. By splitting the above equation in real and imaginary parts it is

$$\left\{\begin{array}{c}|\overline{v}|sin\alpha ={\displaystyle \frac{\sqrt{3}}{2}}\left|V\right|{\displaystyle \frac{t_2}{T_s}}\hfill \\ |\overline{v}|cos\alpha =|V|\left({\displaystyle \frac{t_1}{T_s}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{t_2}{T_s}}\right)\hfill \end{array}\right.$$

(5)

Finally, since $\left|V\right|={\displaystyle \frac{2}{3}}{V}_{dc}$, the times can be written as

$$\left\{\begin{array}{c}{t}_1=\sqrt{3}{\displaystyle \frac{\overline{\left|v\right|}}{V_{dc}}}{T}_{s}sin({\displaystyle \frac{\pi }{3}}-\alpha )\hfill \\ t_2=\sqrt{3}{\displaystyle \frac{\overline{\left|v\right|}}{V_{dc}}}{T}_{s}sin\left(\alpha \right)\hfill \end{array}\right.$$

(6)

3. Methodology

The algorithms have been simulated and then validated with the use of the dSpace MicroLabBox platform. SPM motor parameters are given in Table 1.

3.1. Current Loop Design

Figure 2 and Figure 3 show the block diagrams related to the current control loop in the synchronous reference frame. If the electric motor is isotropic then $L_d=L_q$ [13]. In the control loops the term ${\omega }_m^e{L}_q{i}_q$ and the term ${\omega }_m^e{L}_d{i}_d$ has been added to $u_d$ and $u_q$ respectively, in order to decouple the two dynamics. If the time constant ${\tau }_c$ introduced by the inverter is negligible with respect to the electric time constant of the motor $L_d/R_s$ and $L_q/R_s$, then the terms added to the voltages will cancel those in the motor block diagrams [14]. In order to better control the dynamic of the $i_q$ current loop, also the term ${\omega }_m^e{\mathsf{\Lambda }}_m$ is added to compensate the effect of the back-EMF. The open loop transfer function related to the current loop is then

$$T_{OL}={\displaystyle \frac{k_i+k_{p}s}{s}}{\displaystyle \frac{1}{R_s}}{\displaystyle \frac{1}{1+{\displaystyle \frac{sL}{R_s}}}}{\displaystyle \frac{1}{1+{\displaystyle \frac{s}{{\tau }_c}}}}$$

(7)

The bandwidth is fixed to $f=1kHz$ and the phase margin to ${60}^{\circ }$. The regulator gains that have been calculated from those specifications are $k_p=1.44$ and $k_i=4186$.

3.2. Sensorless Algorithm

The current and voltage are filtered and filtered-integrated by a Second Order - Second order Generalized Integrator (SO-SOGI), which is a forth order adaptive filter, and they are going to be used to estimate ${\mathsf{\Lambda }}_{\alpha }$ and ${\mathsf{\Lambda }}_{\beta }$, which will be used to extract the rotor position. The overall structure is called Second Order Integral Flux Observer (SOIFO) and has strong attenuation capability against the dc offset and harmonics.

3.2.1. Back-emf

The voltage and flux equations of a PMSM motor in $\alpha \beta$ coordinate can be described by the following equations [15].

$${\mathit{u}}_{{\alpha }_{\beta }}=R_s·{\mathit{i}}_{{\alpha }_{\beta }}+{\displaystyle \frac{d}{dt}}{\mathit{\lambda }}_s=R_s·{\mathit{i}}_{{\alpha }_{\beta }}+L_s·{\mathit{i}}_{{\alpha }_{\beta }}+{\mathsf{\Lambda }}_m·\left[\begin{array}{c}cos\left({\theta }_m^e\right)\\ sin\left({\theta }_m^e\right)\end{array}\right]$$

(8)

where ${\mathit{u}}_{{\alpha }_{\beta }}$ is the stator voltage vector, ${\mathit{i}}_{{\alpha }_{\beta }}$ is the stator current vector, ${\lambda }_s$ is the stator flux vector, $L_s$ is the stator inductance, $R_s$ is the stator resistance, ${\theta }_m^e$ is the rotor electrical position and ${\mathsf{\Lambda }}_m$ is the rotor flux linkage. The PMSM sensorless control based on the rotor flux observation is shown in Figure 4, where ${\widehat{\theta }}_m^e$ and ${\widehat{\omega }}_m^e$ are the estimated electromechanical position and the speed.

From (8) the following equation can be obtainted

$${\mathit{\lambda }}_r=\int \left({\mathit{u}}_{\alpha \beta }-R_s{\mathit{i}}_{\alpha \beta }-L_s·{\displaystyle \frac{d}{dt}}{\mathit{i}}_{\alpha \beta }\right)dt$$

(9)

The error due to the integral initial value of the estimated rotor flux is calculated as

$${\mathit{\lambda }}_r\left(0\right)={\mathit{\lambda }}_s\left(0\right)-L{\mathit{i}}_{\alpha \beta }\left(0\right)$$

(10)

On top of that, other errors will be introduced due to parameters mismatch, unknown integral initial value, measurements error, inverter nonlinearities etc. Then the following equation can be obtained

$${\mathit{\lambda }}_r=\int {\mathit{u}}_{\alpha \beta }dt-\int (R_s+\Delta R_s){\mathit{i}}_{\alpha \beta }dt-\int (L_{\alpha \beta }+\Delta L)·{\displaystyle \frac{d}{dt}}{\mathit{i}}_{\alpha \beta }dt+\int {\mathit{e}}_{\alpha \beta 0}dt+\int \chi dt$$

(11)

where $\Delta R_s$ and $\Delta L_s$ are the parameter variation, ${\mathit{e}}_{\alpha \beta 0}$ is the initial back-EMF vector and $\chi$ represents other errors.

3.3. Frequency Locked Loop

The frequency locked loop is a control loop used to auto-adapt the center frequency of the SOGI filter to the fundamental input frequency [16]. Figure 5 shows the architecture of the FLL. The transfer function between the error $\epsilon$ and the input signal is

$$E\left(s\right)={\displaystyle \frac{\epsilon \left(s\right)}{v\left(s\right)}}={\displaystyle \frac{K_1{\omega }^{\prime }s(s^2+{\omega }^{\prime 2})}{[s^4+s^3\left(K_2{\omega }^{\prime }\right)+s^2(2{\omega }^{\prime 2}+K_1{K}_2{\omega }^{\prime 2})+s\left(K_2{\omega }^{\prime 3}\right)+{\omega }^{\prime 4}]}}$$

(12)

The transfer function is a notch filter with unity gain at the centre frequency. The phase angle of this transfer function experiences a phase jump of $+{180}^{\circ }$ when the frequency of the input signal $\omega$ goes from lower to higher to that of the estimated ${\omega }^{\prime }$ of the FLL. Figure 6 shows that the transfer function $E\left(s\right)$ and $x_4^{\prime }\left(s\right)$ are in phase when the frequency of the input signal is lower than that of the FLL ($\omega <{\omega }^{\prime }$), and are in phase opposition otherwise ($\omega >{\omega }^{\prime }$). A new variable ${\epsilon }_f$ is defined as the product between $\epsilon$ and $x_4^{\prime }$. This variable will be positive when $\omega <{\omega }^{\prime }$, 0 when $\omega ={\omega }^{\prime }$ and negative when $\omega >{\omega }^{\prime }$. Based of these considerations, the FLL can be designed using this frequency error ${\epsilon }_f$ as the input to a negative gain $-\gamma$ and an integrator. In this way, if there is a mismatch between the input frequency and the FLL frequency, this control loop will drive the FLL frequency up to the input frequency. Also, in order to accelerate the dynamic, the centre frequency ${\omega }^{\prime }$ can be added as a feed-forward variable, since the magnitude of the input of the FLL decreases as the two frequencies differ, as shown in Figure 6.

3.4. Phase Locked Loop Analysis

Let’s assume the input signal is given by:

$$v=Vsin\left(\theta \right)=Vsin(\omega t+\varphi )$$

(13)

and the output signal is given by:

$$v^{\prime }=cos\left({\theta }^{\prime }\right)=cos({\omega }^{\prime }t+{\varphi }^{\prime })$$

(14)

then the output of the phase multiplier can be written as:

$$\epsilon =Vsin(\omega t+\varphi )cos({\omega }^{\prime }t+{\varphi }^{\prime })={\displaystyle \frac{V}{2}}[\underset{\mathrm{low}\mathrm{frequency}\mathrm{term}}{\underbrace{sin((\omega -{\omega }^{\prime })t+(\varphi -{\varphi }^{\prime }))}}+\underset{\mathrm{high}\mathrm{frequency}\mathrm{term}}{\underbrace{sin((\omega +{\omega }^{\prime })t+(\varphi +{\varphi }^{\prime }))}}]$$

(15)

If the PD regulator of the PLL is a multiplier it is necessary to choose the bandwidth of the system low enough to attenuate the high frequency term, and so only the DC term remains:

$$\epsilon ={\displaystyle \frac{V}{2}}sin((\omega -{\omega }^{\prime })t+(\varphi -{\varphi }^{\prime }))$$

(16)

Assuming that the $V_{CO}$ is tuned to the input frequency, $\omega \approx {\omega }^{\prime }$, then the PD error is further simplified into

$$\epsilon ={\displaystyle \frac{V}{2}}sin(\varphi -{\varphi }^{\prime })$$

(17)

As a last step, assuming that the phase error is low, $\varphi \approx {\varphi }^{\prime }$, the output of the PD regulator can be linearized in the vicinity of that operating point because $sin(\varphi -{\varphi }^{\prime })\approx sin(\theta -{\theta }^{\prime })\approx (\theta -{\theta }^{\prime })$. The result

$$\epsilon ={\displaystyle \frac{V}{2}}(\theta -{\theta }^{\prime })$$

(18)

Assuming V=1, the open loop transfer function in the Laplace domain is

$$F{\left(s\right)}_{OL}=PD\left(s\right)LF\left(s\right)V_{CO}\left(s\right)={\displaystyle \frac{k_p(1+\frac{1}{s{T}_i})}{s^2}}$$

(19)

and the closed loop transfer function is

$$F{\left(s\right)}_{CL}={\displaystyle \frac{s{K}_p+\frac{K_p}{T_i}}{s^2+s{K}_p+\frac{K_p}{T_i}}}$$

(20)

The closed loop transfer function can be normalized in the following way

$$F{\left(s\right)}_{CL}={\displaystyle \frac{2\zeta {\omega }_{n}s+{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}}$$

(21)

where

$${\omega }_n=\sqrt{{\displaystyle \frac{K_p}{T_i}}}\mathrm{and}\xi ={\displaystyle \frac{\sqrt{K_p{T}_i}}{2}}$$

(22)

The response time of this second-order system, from the start of the variation to the 99% of the steady-state final value, can be approximated through

$$t_s=4.6\tau \mathrm{with}\tau ={\displaystyle \frac{1}{\xi {\omega }_n}}$$

(23)

From (20), (21) and (23), the parameters of the controller can be tuned as follows

$$K_p=2\xi {\omega }_n={\displaystyle \frac{9.2}{t_s}}\mathrm{and}{T}_i={\displaystyle \frac{2\xi }{{\omega }_n}}={\displaystyle \frac{t_s{\xi }^2}{2.3}}$$

(24)

3.5. Second Order Integral Flux Observer Architecture

In the SOIFO, there are four SO-SOGI-FLL that have the following quantities as input: $i_{\alpha }$, $i_{\beta }$, $v_{\alpha }$ and $v_{\beta }$. The output of those SO-SOGI-FLL will be the filtered and the integrated filtered input. In this way, the following quantities can be computed

$${\lambda }_{\alpha }=\int (V_{\alpha }-R_s{i}_{\alpha })dt-L_s{i}_{\alpha }$$

(25)

$${\lambda }_{\beta }=\int (V_{\beta }-R_s{i}_{\beta })dt-L_s{i}_{\beta }$$

(26)

Figure 7 show the architecture of the two SOIFO, where ${\widehat{i}}_{\alpha \beta }$ and ${\widehat{v}}_{\alpha \beta }$ represents the filtered variables. ${\lambda }_{\alpha }$ and ${\lambda }_{\beta }$ are used as the input of the QSG-PLL, described hereafter as shown in Figure 8, and the estimated speed ${\widehat{\omega }}_m^e$ and estimated position ${\widehat{\theta }}_m^e$ are obtained.

3.6. Quadrature Signal Generator - Phase Locked Loop

There is a high frequency term in the output of the phase detector. In order to attenuate the effect of this term on the angular velocity and angular position the bandwidth of the control loop must be decreased [17]. A trade-off arises, because a decrease of bandwidth leads to an increase of the settling time. Because of this, a new PLL scheme called Quadrature Signal Generator PLL (QSG-PLL) is introduced in Figure 8. Referring to Figure 8, if $v_{\alpha }=Vcos(\omega t+\varphi )$ and $v_{\beta }=Vsin(\omega t+\varphi )$, the new output of the phase detector can be calculated as

$$\epsilon =Vsin(\omega t+\varphi )cos({\omega }^{\prime }t+{\varphi }^{\prime })-Vcos(\omega t+\varphi )sin({\omega }^{\prime }t+{\varphi }^{\prime })=Vsin((\omega -{\omega }^{\prime }t)+(\varphi -{\varphi }^{\prime }))=Vsin(\theta -\widehat{\theta })$$

(27)

where ${\widehat{\omega }}_m^e$ and ${\widehat{\theta }}_m^e$ are the estimated variables of the PLL. In this way, the high frequency term disappears, and the bandwidth of the control loop can be increased. Also, a normalization block is introduced to make the bandwidth of the control loop independent from the amplitude of the input signals.

4. Results

The electric angular frequency of the back-EMF at which the sensorless algorithm is started has been chosen as ${\omega }_m^e=25\mathrm{rad}/\mathrm{s}$, which is about $48\mathrm{rpm}$. By resorting to the design procedure explained in Section 3.3, the settling time has been chosen as $t_s=0.1s$ and the damping ratio as $\zeta =1/\sqrt{2}$, which leads to an overshoot of $\sigma =4.3\%$. The undamped natural frequency is then $62.23\mathrm{rad}/\mathrm{s}$. Finally, the gains mentioned in Figure 5 are computed as $K_1=1.76$ and $K_2=7.04$.

4.1. Second Order - Second Order Generalized Integrator dynamic

The SO-SOGI is simulated with $v=Acos\left(\omega t\right)$ as input, where A = 1 and ${\omega }_m^e$ = 250 rad/s. Figure 9 and Figure 10 show the results after a step amplitude variation of 20%, and Figure 11 and Figure 12 show the results after a step frequency variation of 20%.

4.2. Second Order - Second Order Generalized Integrator filtering

In this section the filtering capabilites of the sensorless algotithm are described. Figure 13 and Figure 14 show the Bode diagram of variable $x_3\left(s\right)$ and $x_4^{\prime }\left(s\right)$, mentioned in Figure 5 respectively. Figure 13 shows that the slope is -20dB/dec when $\omega \to 0$ and -60dB/dec when $\omega \to \infty$. Instead, Figure 14 shows that the slope is -40dB/dec when $\omega \to 0$ and -40dB/dec when $\omega \to \infty$. The DC gain in the results are almost and not perfectly 0 due to the sampling. In the center frequency the gain is 1 and at high frequency the attenuation is as expected by the Bode diagrams.

4.3. Quadrature Signal Generator - Phase Locked Loop

The time response of the QSG-PLL has been set to $t_r=0.1s$. This constrain leads the parameters of the PLL to be $k_p=92$ and $k_i=4232$. When the motor is accelerating, a static position error will appear while there will not be a static velocity error. That is because of applying a quadratic ramp to a Type two system. The static error can be calculated as $a_e/k_i$, where $a_e$ is the electrical acceleration and $k_i$ is the integral gain of the PI controller. Assuming a maximum mechanical acceleration of $a_m=50rad/s$, which corresponds to electrical acceleration $a_e=250rad/s$ the maximum error will be $0.0035\mathrm{rad}/\mathrm{s}$, that is negligible.

Figure 15 shows the time response of the estimated speed ${\omega }_m^e$ after a frequency step variation of 20%. The time response is about 100ms as designed.

4.4. Steady-State

This section describes the steady-state performance of the sensorless algorithm at

• low speed: ${\omega }_m^e=25\mathrm{rad}/\mathrm{s}$, which is about $47\mathrm{rpm}$
• medium speed: ${\omega }_m^e=250\mathrm{rad}/\mathrm{s}$, which is about $477\mathrm{rpm}$

Figure 16, Figure 17 and Figure 18 show the effect of the speed oscillation due to the cogging torque. Figure 18 and Figure 19 show that the flux linkage ${\lambda }_{\alpha \beta }$ moves along a circle centered in (0,0), because the dc component of the $\alpha$ and $\beta$ emf are effectively rejected. Figure 20 and Figure 21 show the steady state angle error at low and medium speed. The maximum error of angle estimation is 0.25 rad when ${\omega }_m^e$ = 25 rad/s, and 0.12 rad when ${\omega }_m^e$ = 250 rad/s. One part of this error is due to the sampling delay. It can be estimated by $\Delta {\theta }_m^e={\omega }_m^{e}T$ in the worst case, where T is the sampling delay which is $50\mu s$, since $T=1/f$ and $f=20kHz$. At ${\omega }_m^e=25\mathrm{rad}/\mathrm{s}$ the angular error due to the sampling delay is then $\Delta {\theta }_m^e=0.0013\mathrm{rad}$, which is about $0.{0074}^{\circ }$ and at ${\omega }_m^e=250\mathrm{rad}/\mathrm{s}$ is about $0.013\mathrm{rad}/\mathrm{s}$, which is $0.{074}^{\circ }$. The error due to the sampling delay is then negligible at low and medium speed, and the main factor of the angle error is parameter mismatch.

4.5. Dynamic

This section deals with the performance of the sensorless algorithm during a variation of the speed reference and a variation of the load torque. Figure 22 and Figure 23 show the results after a step variation of the ${\omega }_m^e$ reference from 100 to $200\mathrm{rad}/\mathrm{s}$ and vice versa. The results show the stability of the sensorless algorithm even after a fast ${\omega }_m^e$ variation. The maximum angle error is $\Delta {\theta }_m^e=\left|0.7\right|rad$. Figure 24 and Figure 25 show the results after a step load variation of the $+0.4Nm$ and vice versa. The results show the stability of the sensorless algorithm even after a fast torque variation. The maximum angle error is $\Delta {\theta }_m^e=\left|0.5\right|rad$.

5. Conclusions

In this paper, a sensorless algorithm is designed for a PM synchronous motor using an advanced second order-second order generalized integrator filter and a quadrature signal generator phase locked loop controller is used to estimate the rotor position from the filtered output.

The sensorless algorithm is initiated at 25 rad/s electrical angular frequency of the back-emf, that corresponds to 48rpm.

• Low speed: ${\omega }_m^e=25\mathrm{rad}/\mathrm{s}$, approximately $48\mathrm{rpm}$
• Medium speed: ${\omega }_m^e=250\mathrm{rad}/\mathrm{s}$, approximately $477\mathrm{rpm}$

A negligible error of $0.0035\mathrm{rad}$ is recorded at a maximum mechanical acceleration of $a_m=50rad/s$, with $v=Acos\left(\omega t\right)$ as input of SO-SOGI, where $A=1$ and ${\omega }_m^e=250\mathrm{rad}/\mathrm{s}$. The time response of the estimated speed ${\widehat{\omega }}_m^e$ is approximately 100ms after a frequency step variation of 20%. In the steady-state performance of the sensorless algorithm, the maximum error in angle estimation is $0.25\mathrm{rad}$ at ${\omega }_m^e=25\mathrm{rad}/\mathrm{s}$, and $0.12\mathrm{rad}$ at ${\omega }_m^e=250\mathrm{rad}/\mathrm{s}$. Part of this error is due to the sampling delay. It can be estimated by $\Delta {\theta }_m^e={\omega }_m^{e}T$ in the worst case, where T is the sampling delay of $50\mu \mathrm{s}$, given that $T=1/f$ and $f=20\mathrm{kHz}$. At ${\omega }_m^e=25\mathrm{rad}/\mathrm{s}$, the angular error due to the sampling delay is $\Delta {\theta }_m^e=0.0013\mathrm{rad}$, approximately $0.{0074}^{\circ }$, and at ${\omega }_m^e=250\mathrm{rad}/\mathrm{s}$, it is about $0.013\mathrm{rad}$, or $0.{074}^{\circ }$. Thus, the error due to the sampling delay is negligible at both low and medium speeds. Stability of the motor during rapid variation of the reference speed and load torque shows the performance of the sensorless algorithm. The maximum angle error during a fast ${\omega }_m^e$ variation is $\Delta {\theta }_m^e=\pm 0.7\mathrm{rad}$. Similarly, the algorithm remains stable after rapid torque variations, with the maximum angle error of $\Delta {\theta }_m^e=\pm 0.5\mathrm{rad}$.