Sensorless Control of SPM Motor for e-Bike Applications Using Second-Order Integrator Flux Observer

Abstract

The aim of this research is to present both a sensorless control and a torque derating algorithm in the overload region of a permanent magnet motor for e-bikes. First, the theoretical backgrounds and the field-oriented control are presented. Then, a sensorless control is designed based on the back-emf estimation with a second-order generalized integral flux observer for the permanent magnet motor. The second-order generalized integral flux observer is an adaptive filter which can eliminate the DC offset and strongly attenuate the harmonics of the estimated rotor flux. The algorithms have been simulated and then validated by means of tests on a permanent magnet motor for e-bikes.

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]. Due to the absence of excitation currents and having a high power factor, surface-mounted permanent magnet (SPM) synchronous motors are an ideal choice for applications demanding high power efficiency [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 [4,5,6,7]. 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 [8]. However, the mechanical sensors can be replaced with a sensorless algorithm that provides rotor position and speed using the current $I_{dq}$ and reference voltages $V_{dq}$ [9,10,11]. 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 [12].

For medium- and high-speed regions, the most used sensorless algorithms are back-EMF-based, where the rotor position is extracted from the fundamental wave of the back-EMF [13,14,15,16,17]. Back-EMF is calculated from the reference voltage $v_{\alpha \beta }$ and current $i_{\alpha \beta }$. Due to inverter nonlinearity, dead time, parameter variations, and measurement errors, the measured current contains harmonics and unwanted frequency components [11,18,19]. Several research articles have been published that use LC filters, sliding-mode filters, generalized extended state observers, dual sliding-mode observers, modified sliding-mode observers, and adaptive observers to improve estimation by filtering the measured current and voltage [15,20,21,22,23]. In this paper, a second-order generalized integrator is used as a filter. From the results, it is concluded that the proposed filtering algorithm improves sensorless position estimation.

In this paper, the mathematical equations of the SPM motor in the rotating reference frame are first described in Section 2.1. Then, the FOC system and space vector modulation (SVM) technique with sensors are explained in Section 2.2 and Section 2.3, respectively. In Section 3.1, a current loop controller with a sensorless algorithm is designed for the SPM motor based on the given specifications. The back-EMF is estimated in Section Back-Emf using the frequency-locked loop explained in Section 3.3 and phase-locked loop analysis explained in Section 3.4 with a Second-Order Integral Flux Observer (SOIFO) filter given in Section 3.5. Finally, in Section 3.6, the 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 in Section 5.

2. Standard Control of SPM Motor

2.1. Surface-Mounted Permanent Magnet Motor

The surface-mounted permanent magnet (SPM) synchronous motor is described by the following equations in the rotating reference frame [24,25,26,27,28,29]:

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

(1)

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

(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, ${\Lambda }_m$ is the flux linkage produced by the magnets, $R_s$ is the phase resistance, and $L_d$ and $L_q$ are the d-axis and q-axis synchronous inductances; in addition, $L_d=L_q$ in the SPM motor.

2.2. Field-Oriented Control

FOC is the most used control for SPM motors. 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 [30]. Electromagnetic torque is a function of the angle between the stator and rotor flux. The equation that describes the induced electromagnetic torque is [31,32,33,34,35]

$${\tau }_{em}={\displaystyle \frac{3}{2}}p{\Lambda }_m{i}_q$$

(3)

where $i_q$ is the quadrature current, and p is the number of pole pairs. Such a torque has to balance the torque at the shaft, which includes the load torque ${\tau }_L$, friction torque $B{\omega }_m$, and the acceleration torque $J{\displaystyle \frac{d{\omega }_m}{dt}}$ and is given as

$${\tau }_{em}={\tau }_L+B{\omega }_m+J{\displaystyle \frac{d{\omega }_m}{dt}}$$

(4)

where B is the viscous friction coefficient, ${\omega }_m$ is the mechanical speed, and J is the inertia.

Figure 1 shows the FOC architecture. The three-phase currents and the rotor angle are measured on motor terminals and the rotor shaft, respectively. The three-phase currents are converted into a two-phase vector frame, $i_{\alpha }$ and $i_{\beta }$, in the stationary state, using the Clarke transformation, and then to the rotating reference frame using the Park transformation [36]. 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 [37]. 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 a 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 of controlling the stator currents and the torque. The idea behind the operation is the transformation from a three-phase system into a two-coordinate system [38].

2.3. Space Vector Modulation

SVM is a modulation technique able to generate the desired phase voltages [39]. Since only one mosfet is ON and the other is OFF in each leg, there are eight possible combinations. The phase vectors divide the plan into six 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 [40,41,42,43,44,45]. Let us assume the desired voltage is in the first sector (0°–60°); the following condition holds:

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

(5)

where $t_1$ and $t_2$ are the times in which the vectors ${\overline{v}}_{100}$ and ${\overline{v}}_{110}$ are applied, respectively, and $t_0$ is the time in which the zero vector ${\overline{v}}_{111}$ is applied. Splitting the above equation into real and imaginary parts yields

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

(6)

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

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

(7)

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. Motor parameters.

Parameter	Symbol	Value	Unit
Pole number	$2p$	10	-
PM flux linkage	${\Lambda }_m$	0.0144	Vs
Winding resistance	$R_s$	0.222	Ω
Winding inductance	$L_s$	0.25	mH
Nominal speed	$n_n$	2500	rpm
Nominal torque	$T_n$	2	Nm

, and the cross-section in shown in Figure 2.

3.1. Current Loop Design

Figure 3 and Figure 4 show the block diagrams related to the current control loop in the synchronous reference frame. If the electric motor is isotropic, then $L_d=L_q$ [46]. In the control loops, the term ${\omega }_m^e{L}_q{i}_q$ and the term ${\omega }_m^e{L}_d{i}_d$ have been added to $v_d$ and $v_q$, 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 [47]. In order to better control the dynamic of the $i_q$ current loop, the term ${\omega }_m^e{\Lambda }_m$ is also added to compensate for the effect of the back-EMF. The open-loop transfer function related to the current loop is then

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

(8)

The bandwidth is fixed at $f=1 \mathrm{kHz}$ and the phase margin at ${60}^{\circ }$. The regulator gains that have been calculated from these 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 ${\Lambda }_{\alpha }$ and ${\Lambda }_{\beta }$, which will be used to extract the rotor position. The overall structure is called the Second-Order Integral Flux Observer (SOIFO) and has strong attenuation capability against the dc offset and harmonics.

Back-Emf

The voltage and flux equations of an SPM motor in $\alpha \beta$ coordinates can be described by the following equations [48]:

$$\begin{array}{cc}\hfill {\mathit{v}}_{{\mathit{\alpha }}_{\mathit{\beta }}}& =R_s·{\mathit{i}}_{{\mathit{\alpha }}_{\mathit{\beta }}}+{\displaystyle \frac{d}{dt}}{\mathit{\lambda }}_{\mathit{s}}\hfill \\ \hfill {\mathit{v}}_{{\mathit{\alpha }}_{\mathit{\beta }}}& =R_s·{\mathit{i}}_{{\mathit{\alpha }}_{\mathit{\beta }}}+{\displaystyle \frac{d}{dt}}\left(L_s·{\mathit{i}}_{{\mathit{\alpha }}_{\mathit{\beta }}}+{\mathbf{\Lambda }}_{\mathit{m}}·\left[\begin{array}{c}cos\left({\theta }_m^e\right)\\ sin\left({\theta }_m^e\right)\end{array}\right]\right)\hfill \end{array}$$

(9)

where ${\mathit{v}}_{{\mathit{\alpha }}_{\mathit{\beta }}}$ is the stator voltage vector, ${\mathit{i}}_{{\mathit{\alpha }}_{\mathit{\beta }}}$ is the stator current vector, ${\mathit{\lambda }}_{\mathit{s}}$ 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 ${\mathbf{\Lambda }}_{\mathit{m}}$ is the rotor flux linkage. The SPM sensorless control based on the rotor flux observation is shown in Figure 5, where ${\widehat{\theta }}_m^e$ and ${\widehat{\omega }}_m^e$ are the estimated electromechanical position and the speed.

From (9), the following equation can be obtained:

$${\mathit{\lambda }}_{\mathit{r}}\left(t\right)={\int }_{t_0}^t\left({\mathit{v}}_{\mathit{\alpha }\mathit{\beta }}\left(t\right)-R_s{\mathit{i}}_{\mathit{\alpha }\mathit{\beta }}\left(t\right)-L_s·{\displaystyle \frac{d}{dt}}{\mathit{i}}_{\mathit{\alpha }\mathit{\beta }}\left(t\right)\right)dt$$

(10)

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

$${\mathit{\lambda }}_{\mathit{r}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}={\mathit{\lambda }}_{\mathit{s}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}-L{\mathit{i}}_{\mathit{\alpha }\mathit{\beta }}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$$

(11)

On top of that, other errors will be introduced due to parameter mismatch, unknown integral initial values, measurement error, inverter nonlinearities, etc. Then, the following equation can be obtained:

$$\begin{array}{cc}\hfill {\mathit{\lambda }}_{\mathit{r}}\left(t\right)=& {\int }_{t_0}^t{\mathit{v}}_{\mathit{\alpha }\mathit{\beta }}\left(t\right)dt-{\int }_{t_0}^t(R_s+\Delta R_s){\mathit{i}}_{\mathit{\alpha }\mathit{\beta }}\left(t\right)dt-{\int }_{t_0}^t(L_{\alpha \beta }\left(t\right)+\Delta L)·{\displaystyle \frac{d}{dt}}{\mathit{i}}_{\mathit{\alpha }\mathit{\beta }}\left(t\right)dt\hfill \\ \hfill & +{\int }_{t_0}^t{\mathit{e}}_{\mathit{\alpha }\mathit{\beta }\mathbf{0}}\left(t\right)dt+{\int }_{t_0}^t\chi \left(t\right)dt\hfill \end{array}$$

(12)

where $\Delta R_s$ and $\Delta L_s$ are the parameter variation, ${\mathit{e}}_{\mathit{\alpha }\mathit{\beta }\mathbf{0}}$ 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 [49]. Figure 6 shows the architecture of the FLL. The transfer function between the error $\epsilon$ and the input signal is

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

(13)

In Section 3.3 and in Figure 6, V is the input voltage, ω′ is the input signal frequency, $X_1$ and $X_2$ are the filtered outputs, $K_1$ and $K_2$ are the gain coefficients, $X_3$ is the final filtered output, $X_4$ is the integrated–filtered output, and $ϵ$ is the error signal. The transfer function is a notch filter with unity gain at the center 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 than that of the estimated ω′ of the FLL. Figure 7 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 on 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 center 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 7.

3.4. Phase-Locked Loop Analysis

Assuming that the input signal is given by

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

(14)

and the output signal is given by

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

(15)

the output of the phase multiplier can be written as

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

(16)

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

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

(17)

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

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

(18)

As a last step, assuming that the phase error is low, $\varphi \approx {\varphi }^{\prime }$, the output of the PI 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 is

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

(19)

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

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

(20)

and the closed-loop transfer function is

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

(21)

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

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

(22)

where

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

(23)

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

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

(24)

From (21), (22), and (24), the parameters of the controller can be tuned as follows:

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

(25)

3.5. Second-Order Integral Flux Observer Architecture

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

$${\lambda }_{\alpha }\left(t\right)={\int }_{t_0}^t(v_{\alpha }\left(t\right)-R_s{i}_{\alpha }\left(t\right))dt-L_s{i}_{\alpha }\left(t\right)$$

(26)

$${\lambda }_{\beta }\left(t\right)={\int }_{t_0}^t(v_{\beta }-R_s{i}_{\beta }\left(t\right))dt-L_s{i}_{\beta }\left(t\right)$$

(27)

Figure 8 shows the architecture of the two SOIFOs, where ${\widehat{i}}_{\alpha \beta }$ and ${\widehat{v}}_{\alpha \beta }$ represent the filtered variables. ${\lambda }_{\alpha }$ and ${\lambda }_{\beta }$ are used as the input of the QSG-PLL, described hereafter as shown in Figure 9, and the estimated speed ${\widehat{\omega }}_m^e$ and estimated position ${\widehat{\theta }}_m^e$ are obtained.

3.6. Quadrature Signal Generator (QSG)–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 [50]. A trade-off arises, because a decrease in bandwidth leads to an increase in the settling time. Because of this, a new PLL scheme called the quadrature signal generator–PLL (QSG-PLL) is introduced in Figure 9. Referring to Figure 9, 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

$$\begin{array}{cc}\hfill \epsilon & =Vsin(\omega t+\varphi )cos({\omega }^{\prime }t+{\varphi }^{\prime })-Vcos(\omega t+\varphi )sin({\omega }^{\prime }t+{\varphi }^{\prime })\hfill \\ \hfill \epsilon & =Vsin((\omega -{\omega }^{\prime }t)+(\varphi -{\varphi }^{\prime }))\hfill \\ \hfill \epsilon & =Vsin(\theta -\widehat{\theta })\hfill \end{array}$$

(28)

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.

Figure 9. QSG-PLL block diagram.

Figure 9. QSG-PLL block diagram.

4. Experimental Results

The proposed algorithms are implemented on a test bench for the SPM machine described in Table 1 using the dSpace MicroLabBox platform. In this platform, there is one Microcontroller (MCU) and one Field-Programmable Gate Array (FPGA). However, only the MCU has been used in this work. Multiple ADCs and DACs are available through BNC connectors, with a resolution of 14 to 16 bits. A large number of digital I/O pins can be accessed via two 50-pin serial cables. Additional interfaces, such as USB and CAN, are also present but were not utilized. The algorithm was developed in the Simulink© environment and subsequently compiled into C code for execution on the MCU. The dSPACE software, ControlDesk, is employed to communicate with the MCU in real time, allowing monitoring of the algorithm’s status.

In the experimental test, the electric angular frequency of the back-EMF at which the proposed 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 $t_s=0.1$ s 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 6 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 10 and Figure 12 show the results after a step amplitude variation of 20%, and Figure 11 and Figure 13 show the results after a step frequency variation of 20%.

4.2. Second-Order–Second-Order Generalized Integrator Filtering

In this section, the filtering capabilities of the sensorless algorithm are described. Figure 14 and Figure 15 show the Bode diagram of variables $x_3\left(s\right)$ and $x_4^{\prime }\left(s\right)$, mentioned in Figure 6, respectively. Figure 14 shows that the slope is −20 dB/dec when $\omega \to 0$ and −60 dB/dec when $\omega \to \infty$. Instead, Figure 15 shows that the slope is −40 dB/dec when $\omega \to 0$ and −40 dB/dec when $\omega \to \infty$. The DC gain in the results is almost but not exactly 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 $t_r=0.1$ s. This constraint 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 $a_m=50$ rad/s, which corresponds to electrical acceleration of $a_e=250$ rad/s, the maximum error will be $0.0035\mathrm{rad}/\mathrm{s}$, which is negligible.

Figure 16 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 17, Figure 18 and Figure 19 show the effect of the speed oscillation due to the cogging torque. Figure 19 and Figure 20 show that the flux linkage ${\lambda }_{\alpha \beta }$ moves along a circle centered at (0,0), because the dc components of the $\alpha$ and $\beta$ emf are effectively rejected. Figure 21 and Figure 22 show the steady-state angle error at low and medium speeds. 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 μs, since $T=1/f$ and $f=20$ kHz. 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 ${\omega }_m^e=250\mathrm{rad}/\mathrm{s}$, it 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 speeds, 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 in the reference speed and a variation in the load torque. Figure 23 and Figure 24 show the results after a step variation in 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 25 and Figure 26 show the results after a step load variation of $+0.4$ Nm 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 an SPM 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 a 25 rad/s electrical angular frequency of the back-emf, which corresponds to 48 rpm, and also verified at medium speed: ${\omega }_m^e=250\mathrm{rad}/\mathrm{s}$, approximately 477 rpm. A negligible error of $0.0035\mathrm{rad}$ is recorded at a maximum mechanical acceleration of $a_m=50$ rad/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 μs, given that $T=1/f$ and $f=20$ 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 in 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 a maximum angle error of $\Delta {\theta }_m^e=\pm 0.5\mathrm{rad}$.