An Optimized Time Sequence for Sensorless Control of IPMSM Drives via High-Frequency Square-Wave Signal Injection Scheme

Abstract

This paper presents a filterless sensorless control scheme with an optimized time sequence based on high-frequency (HF) square-wave voltage injection for a five-phase interior permanent magnet machine (IPMSM) drive. To avoid the utilization of low-pass filters (LPFs) in signal processing, an effective method without filters is proposed in this paper. Moreover, the cross-coupling magnetic saturation is analyzed and the online position error compensation is applied based on the offline measurements and finite-element analysis (FEA). Besides, compared with the conventional time sequence of senseorless control, the proposed optimized time sequence can eliminate the additional position estimation error caused by the time delay in digital implementation. Numerical simulations and experiments with a 2-kW five-phase IPMSM are carried out. The results verify the feasibility and effectiveness of the proposed sensorless control scheme with an optimized time sequence adopted by the IPMSM drives.

1. Introduction

Because of high torque density, excellent dynamic performance and wide adjustment speed range, the interior permanent magnet synchronous motors (IPMSMs) have been extensively employed in modern industrial applications, including aerospace aircraft, electric vehicles and hybrid renewable power systems [1,2,3,4]. In these applications, the position sensor for IPMSM drives, which fails easily in rigorous working environments, can be eliminated by sensorless control scheme considering cost reduction and reliability enhancement.

Tremendous effort has been invested in sensorless control in the last few decades. Generally, the sensorless control methods can be mainly classified into the following two categories: fundamental excitation based methods and external signal injection excitation based methods The former exploits the back electromotive force (EMF) or flux linkage to extract and estimate the rotor position information [5]. These methods are widely applied in medium and higher speed regions, while they fail at the standstill and low speed causes due to the small signal-to-noise ratio (SNR) of EMF signal [6]. To expand the sensorless control into the standstill and low speed range, the latter are adopted to obtain the rotor position information by external signal injection excitation. In terms of the types of injected signals, the mainstream signal injection methods can be roughly categorized as follow:

(1)

Use low-frequency signal injection [7,8,9].

(2)

Use HF rotational sinusoidal voltage injection (RSVI) [10,11].

(3)

Use HF pulsing signal injection such as sinusoidal voltage injection (SVI) [12,13,14] or square-wave voltage injection (SWVI) [15,16,17,18,19,20,21,22,23,24].

The first kind of method can effectively reduce the audible noise caused by the additional HF signal. However, the accuracy of position estimation will significantly decrease due to the low injected frequency, which need some additional auxiliary measures to enhance the estimation performance. The second kind of method uses HF sinusoidal voltages to inject into the stationary reference frame (SRF), which can cause inevitable torque vibration due to the fluctuations in the q-axis current [18]. For the third type of method, similar to the RSVI method, the application of the low-pass filters (LPFs) in HF pulsing SVI method limits the dynamic bandwidth of the system. In [19], the bandwidth of the speed loop is limited to a few hertz with sinusoidal signal injection, which makes it difficult to meet the requirements of high-performance drive. Moreover, the highest frequency of the injected sinusoidal signal is limited to about 1/10 of the carrier wave frequency [20]. Different from RSVI and pulsing SVI, the injection frequency of SWVI method can be increased up to the carrier frequency. Besides, the cutoff frequency of LPFs for fundamental frequency (FF) current extraction in SWVI method can be set higher, which effectively enhances the bandwidth of the drive system. For these reasons, the HF SWVI method can be considered as the optimal option for sensorless IPMSM drives in low speed regions. Nevertheless, the accuracy of position estimation in the HF SWVI method is still insufficient since it suffers from the position angle error caused by the cross-coupling magnetic saturation, inverter nonlinearity, and digital time-delay effect [21,22,23,24].

Numerous studies have been implemented to address these problems. To compensate for the position errors due to the cross-coupling effects, FEA [25], structured neural network, and offline measurements [26,27,28] are adopted. In [21,22] and [29], the nonlinearity effects of the inverter are analyzed, and related compensation measures are also proposed. However, there is a relatively small body of literature that is concerned with the digital time-delay in position sensorless control.

The two main contributions of this paper can be summarized as follows:

(1)

To avoid the utilization of LPFs in signal processing, the filterless signal processing method is proposed, which can enhance the system bandwidth.

(2)

The digital time delay in sensorless control is discussed, and an optimized time sequence is proposed, which can eliminate the additional position estimation error caused by the digital time delay.

Besides, to attenuate the effect of cross-coupling magnetic saturation, the online position error compensation is adopted based on FEA and offline measurements. Both simulations and experiment results are illustrated to demonstrate the effectiveness of the proposed method.

The rest of this paper is outlined as follows. In Section 2, the basic principles of the filterless HF SWVI method for a five-phase IPMSM drive are analyzed. In Section 3, the digital time delay and cross-coupling magnetic saturation in sensorless control is discussed, and rated compensation strategies are also adopted to enhance the accuracy of position estimation. In Section 4, the proposed sensorless scheme is verified by both simulation and experimental results. Finally, conclusions are given in Section 5.

2. Filterless HF SWVI Scheme for IPMSM Drive

2.1. Mechanism of HF SWVI Scheme

To facilitate the control of the five-phase IPMSM, the natural coordinate frame ABCDE can be decomposed into three mutually orthogonal spaces by means of coordinate transformation: fundamental space d–q, third harmonic space d₃–q₃, and zero-sequence space [1]. The first two spaces are the synchronous rotating reference frame (RRF) with different rotational speeds. Besides, the third harmonic component does not participate in the electromechanical energy conversion. Meanwhile, due to the star connection of the stator winding, the neutral point current in the zero-sequence space is zero, which can be neglected. Consequently, the stator voltage equation of the five-phase IPMSM can be demonstrated as

$$\left[\begin{array}{c}{u}_d\\ u_q\\ u_{d3}\\ u_{q3}\end{array}\right]=\left[\begin{array}{cccc}{R}_s+p{L}_d& -{\omega }_e{L}_q& 0& 0\\ {\omega }_e{L}_d& R_s+p{L}_q& 0& 0\\ 0& 0& R_s+p{L}_{d3}& -3{\omega }_e{L}_{q3}\\ 0& 0& 3{\omega }_e{L}_{d1}& R_s+p{L}_{q3}\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\\ i_{d3}\\ i_{q3}\end{array}\right]+{\omega }_e\left[\begin{array}{c}0\\ {\psi }_f\\ 0\\ 3{\psi }_{f3}\end{array}\right]$$

(1)

where u_d,q and i_d,q are the voltage and current vectors, respectively; R_s is the stator resistance; L_d and L_q are the inductances on the d-axis and q-axis, respectively; ω_e is the electrical angular speed; Ψ_f is the flux linkage; p = d/dt is a differential operator of time. The subscript “3” represents the corresponding components in the third harmonic spaces. It is worth noting that the HF SWVI method only acts on the fundamental space d-q and does not involve the third harmonic space d₃–q₃.

Since the injection frequency is much higher than the fundamental frequency and the motor operates at zero- and low-speed region, (1) can be simply depicted as

$$\left[\begin{array}{c}{u}_{dh}\\ u_{qh}\end{array}\right]=\left[\begin{array}{cc}{L}_d& 0\\ 0& L_q\end{array}\right]\frac{d}{dt}\left[\begin{array}{c}{i}_{dh}\\ i_{qh}\end{array}\right]$$

(2)

where the subscript “h” is the corresponding HF component.

Then, the currents variation due to the injected voltage in the estimated rotating frame (ERF) can be shown as

$$\frac{d}{dt}\left[\begin{array}{c}{\widehat{i}}_{dh}\\ {\widehat{i}}_{qh}\end{array}\right]=\frac{1}{2L_d{L}_q}\left[\begin{array}{cc}{L}_{\mathsf{\Sigma }}+L_{dif}\cos \left(2{\theta }_{err}\right)& L_{dif}\sin \left(2{\theta }_{err}\right)\\ L_{dif}\sin \left(2{\theta }_{err}\right)& L_{\mathsf{\Sigma }}-L_{dif}\cos \left(2{\theta }_{err}\right)\end{array}\right]\left[\begin{array}{c}{\widehat{u}}_{dh}\\ {\widehat{u}}_{qh}\end{array}\right]$$

(3)

where the superscript “^” is the corresponding component in the ERF. L_Σ and L_dif are defined as L_Σ ≡ L_d + L_q and L_dif ≡ L_q − L_d, respectively. θ_err is the rotor position estimation error, which is defined as the difference between the actual position θ_e and the estimated position $\widehat{\theta }$.

The HF square-wave voltage signal injected into the ERF can be expressed as

$$u_h=\left\{\begin{array}{cc}{U}_h& t\in \left(0,\frac{T_h}{2}\right)\\ -U_h& t\in \left(\frac{T_h}{2},T_h\right)\end{array}\right.$$

(4)

where T_h is the period of HF voltage signal. After substituting (4) into (3), and de-symbolizing the envelope curves of induced HF current, the amplitude of the induced HF current in the ERF can be obtained as

$$\left[\begin{array}{c}{\widehat{I}}_{dh}\\ {\widehat{I}}_{qh}\end{array}\right]=\frac{T_h·U_h}{8L_d{L}_q}\left[\begin{array}{c}{L}_{\mathsf{\Sigma }}+L_{dif}\cos \left(2{\theta }_{err}\right)\\ L_{dif}\sin \left(2{\theta }_{err}\right)\end{array}\right]$$

(5)

As can be seen, ${\widehat{I}}_{qh}$ in (5) contains the position error information. Hence, when this position error is sufficiently small and L_dif ≠ 0, the error signal can be linearized as (6) after normalization.

$$f\left({\theta }_{err}\right)={\widehat{I}}_{qh}·K_{norm}=\frac{\sin \left(2{\theta }_{err}\right)}{2}\approx {\theta }_{err}$$

(6)

where K_norm is the normalization coefficient and can be presented as

$$K_{norm}=\frac{4L_d{L}_q}{T_h{U}_h{L}_{dif}}$$

(7)

Then, a PI-type phase-locked loop (PLL) in Figure 1 can be applied to estimate the rotor position and speed.

2.2. Proposed Filterless Signal Processing Method

There are some certain drawbacks associated with the application of filters in signal processing, e.g., an inevitable processing delay, bandwidth limitation, and performance qualities diminishment. Hence, in this paper, a signal processing method without filters is proposed. Figure 2 illustrates the proposed filterless signal processing method based on HF SWVI scheme. In Figure 2, the nearest four vectors SVPWM (NFV-PWM) [30] is applied to the SVPWM control of five-phase inverter, which can effectively suppress the third harmonic. The FF current is obtained by sampling at zero-crossing point, meanwhile the induced HF current of q-axis is extracted by simple algebraic operation.

Figure 3 illustrates the time sequence of the injected square-wave voltage and the induced HF current, which shows the details of the proposed signal processing methods. In Figure 3, the induced HF current is sampled four times during one HF square-wave voltage period. Besides, the induced HF current has a fixed zero-crossing point; therefore, the sampling current obtained at the zero-crossing point is the FF current. Since the frequency of the injected square-wave signal is much higher than the frequency of FF current, the growth rate of FF current in the one carrier wave period (T_c) can be regarded as constant [31,32], while the high-frequency current growth rate is reversed. Besides, the update frequency of FF voltage is set to be twice the HF injected signals frequency. According to the above analysis, the currents at different sampling instant can be expressed as

$$\left\{\begin{array}{c}{\widehat{i}}_{d,qb{k}_0}={\widehat{i}}_{d,qs{k}_0},{\widehat{i}}_{d,qb{k}_2}={\widehat{i}}_{d,qs{k}_2}\\ {\widehat{i}}_{d,qh{k}_1}={\widehat{i}}_{d,qs{k}_1}-\frac{{\widehat{i}}_{d,qs{k}_0}+{\widehat{i}}_{d,qs{k}_2}}{2}\end{array}\right.$$

(8)

where the subscript “b” is the corresponding FF component, the subscript “s” is the corresponding sampling component, and the subscript “k_i” means the corresponding component at k_i sampling instant (i = 0,1,2…). For any sampling instant, the sampling current at d-or q-axis ${\widehat{i}}_{d,qs{k}_i}$ contains the FF current ${\widehat{i}}_{d,qb{k}_i}$ and the induced HF current ${\widehat{i}}_{d,qh{k}_i}$.

The obtained FF current can be directly used for the current loop without passing through a low-pass filter or notch filter. Furthermore, the induced HF current is used to observing rotor position and velocity. The filterless signal processing method is suitable for digital control, which can achieve high performance during sensorless operation.

3. Analysis and Compensation of Position Estimation Error

Since the digital time delay and cross-coupling magnetic saturation will affect the position estimation accuracy of sensorless control, we will give the corresponding analysis and compensation strategies for these two problems in this section.

3.1. Analysis of Digital Time Delay Effect

In full-digital control system of IPMSM drives, there is an inevitable execution time delay, which is commonly one sampling period. Furthermore, the digital PWM logic also produces a further time delay [33]. The delay caused by the PWM is relevant to the strategy of generating PWM signals in a digital way. Figure 4a–c illustrates a digital PWM output mode, which is called the double current sampling and double PWM duty cycle update (DSDU) mode. Under this mode, the update frequency of duty cycle is set to be twice the carrier wave frequency, and one sampling time delay T_s is a good approximation for implementing digital PWM logic. This two sampling period time delay (2T_s or 1T_c) cannot be neglected when considering the frequency of injected HF signal is close to the PWM switching frequency. Meanwhile, in realization, the actual injected HF voltage lags behind the HF signal reference, which will produce an additional position estimation error [29]. Figure 4d illustrates the HF voltage reference. The actual HF voltage and induced current considering one carrier cycle time delay are shown in Figure 4e,f. In this part, the position estimation error caused by digital time delay is discussed and a compensation scheme based on optimized time sequence is proposed.

Figure 5 illustrates the conventional time sequence for sensorless control based on the DSDU mode. After each PWM interrupt is triggered, the DSP will sequentially execute the operations of I~V. In every sampling period, the coordinate transformation of sampling current is based on the estimated position angle, which is obtained in the previous sampling period. The subscript “m” is the corresponding position angle in the m th sampling instant.

During the sampling instant (m−1, m), the HF voltage of the inverter output in the real RRF can be expressed as

$${\left[\begin{array}{c}{u}_{dh}\\ u_{qh}\end{array}\right]}_{<m-1,m>}=T_{Park}\left({\theta }_e(t)\right)·T_{Park}^{-1}\left({\widehat{\theta }}_{m-2}\right)·\left[\begin{array}{c}{U}_h\\ 0\end{array}\right]$$

(9)

where the T_Park and ${\mathrm{T}}_{\mathrm{Park}}^{-1}$ are the Park coordinate transformation and its inverse transformation respectively.

In (9), since the θ_e(t) varies from (m−1) to (m) sampling instant, the HF voltage in the real RRF also varies with time, and the induced HF current amplitude at the (m) sampling instant can be shown as

$${\left[\begin{array}{c}{I}_{dh}\\ I_{qh}\end{array}\right]}_{<m>}=\underset{m-1}{\overset{m}{\int }}\left[\begin{array}{cc}\frac{1}{L_d}& 0\\ 0& \frac{1}{L_d}\end{array}\right]{\left[\begin{array}{c}{u}_{dh}\\ u_{qh}\end{array}\right]}_{<m-1,m>}dt$$

(10)

Consider that θ_e(t) changes very slowly during the (m−1) to (m) sampling period, which can be replaced by its average value ${\overline{\theta }}_{e,m}$ for the integral in (10). Here, ${\overline{\theta }}_{e,m}={\theta }_{e,m-1}+0.5{\omega }_e{T}_s$ and the HF current amplitude in (14) can be deduced as

$${\left[\begin{array}{c}{I}_{dh}\\ I_{qh}\end{array}\right]}_{<m>}=\left[\begin{array}{c}\frac{1}{L_d}\cos \left({\overline{\theta }}_{e,m}-{\widehat{\theta }}_{m-2}\right)\\ -\frac{1}{L_q}\sin \left({\overline{\theta }}_{e,m}-{\widehat{\theta }}_{m-2}\right)\end{array}\right]U_h\cdot T_s$$

(11)

Then the amplitude of the HF current in the ERF at the (m) sampling instant can be deduced as

$${\left[\begin{array}{c}{\widehat{I}}_{dh}\\ {\widehat{I}}_{qh}\end{array}\right]}_{<m>}=T_{Park}\left({\widehat{\theta }}_{m-1}\right)T_{Park}^{-1}({\theta }_{e,m})\cdot {\left[\begin{array}{c}{I}_{dh}\\ I_{qh}\end{array}\right]}_{<m>}$$

(12)

where θ_e,m is the real electrical angle at (m) sampling instant. Therefore, the amplitude of q-axis HF current in the ERF can be shown in (13).

$${\widehat{I}}_{qh,m}=\frac{T_s{U}_h}{2L_d{L}_q}[-\stackrel{\mathrm{digital}\mathrm{time}-\mathrm{delay}\mathrm{error}}{\overbrace{L_{\mathsf{\Sigma }}\sin \left({\widehat{\theta }}_{m-1}-{\widehat{\theta }}_{m-2}-0.5{\omega }_e{T}_s\right)}}+L_{dif}\sin \left(2{\theta }_{e,m}-{\widehat{\theta }}_{m-1}-{\widehat{\theta }}_{m-2}-0.5{\omega }_e{T}_s\right)]$$

(13)

Compared with (5), the coefficient of the L_Σ term in (13) does not equal to zero, which can be defined as the digital time-delay error. When the PLL converges to zero steadily, i.e., ${\widehat{I}}_{qh}=0$, the rotor position estimation error can be expressed as

$${\theta }_{err}\approx \frac{{\omega }_e{T}_s}{4}\left(\frac{L_{\mathsf{\Sigma }}}{L_{dif}}-5\right)=\frac{{\omega }_e{T}_s}{4}\left(\frac{2}{\lambda -1}-4\right)$$

(14)

where λ = L_q/L_d is the saliency rate. The digital time-delay error in (13) brings the steady-state error, which is proportional to the speed and inversely proportional to the saliency rate.

3.2. Compensation Strategy for Digital Time-Delay

As can be seen from (13) and (14), the digital time-delay error is dependent on the motor parameters and speed, which is difficult to be directly compensated. In order to attenuate the time-delay effect, an optimized time sequence is proposed in Figure 6. Compared with the conventional time sequence, the optimized time sequence compensates the estimated angle in the step II, i.e., for any (m) sampling instant, the sampling current coordinate transformation is based on

$${\theta }_m={\widehat{\theta }}_{m-1}+\frac{{\widehat{\omega }}_e{T}_s}{2}$$

(15)

For (m−1) sampling instant, (15) can be reformulated as

$${\theta }_{m-1}={\widehat{\theta }}_{m-2}+\frac{{\widehat{\omega }}_e{T}_s}{2}$$

(16)

Therefore, substituting (16) into (13), the HF current in the ERF can be expressed as

$${\widehat{I}}_{qh,m}=\frac{T_s{U}_h}{2L_d{L}_q}{L}_{dif}\sin \left(2{\theta }_{e,m}-{\widehat{\theta }}_{m-1}-{\widehat{\theta }}_{m-2}-0.5{\omega }_e{T}_s\right)$$

(17)

As shown in (17), the digital time-delay error in (13) is eliminated. According to the optimized time sequence, the position estimation error in (17) when the PLL is steady-state convergence can be obtained as

$${\theta }_{err}=-1.5{\omega }_e{T}_s$$

(18)

As is shown in (18), the observed position estimation error is negative, which indicates that the advanced estimation is realized.

3.3. Compensation for Cross-Coupling Magnetic Saturation

The cross-coupling magnetic saturation between the d- and q-axis can significantly influence the accuracy of rotor-position estimation in sensorless control. When the cross-coupling magnetic saturation is considered, (2) can be rewritten as

$$\left[\begin{array}{c}{u}_{dh}\\ u_{qh}\end{array}\right]=\left[\begin{array}{cc}{L}_d& L_{dq}\\ L_{dq}& L_q\end{array}\right]\frac{d}{dt}\left[\begin{array}{c}{i}_{dh}\\ i_{qh}\end{array}\right]$$

(19)

where L_dq is the cross-coupling inductance. Therefore, the amplitude of the HF q-axis current in the ERF can be expressed as [26]

$${\widehat{I}}_{qh}=\frac{T_s\cdot U_h\sqrt{L_{dif}^2+L_{dq}^2}}{2\left(L_d{L}_q-L_{dq}^2\right)}·\sin \left(2{\theta }_{err}-{\phi }_{dq}\right)$$

(20)

where φ_dq = tan⁻¹ (L_dq/L_dif). Hence, the position estimation error caused by the cross-coupling inductance can be given as

$${\theta }_{err}=\frac{{\phi }_{dq}}{2}$$

(21)

As can be seen, the position estimation error contains a phase shift of 0.5φ_dq. Figure 7 shows the finite element analysis (FEA) results of L_dif, L_dq and θ_err under different d- and q-axis current condition. As can be seen from Figure 7c, the position estimation error caused by cross-coupling magnetic saturation can increase with the amplitude of the q-axis current.

To further accurately eliminate the effect of cross-coupling magnetic saturation, in this paper, the offline measurement is applied to eliminate the additional position estimation error. For the prototype machine, the offline measured position estimation error angle when I_d = 0 is shown in Figure 8. Compared with the FEA results, the fitting cure obtained by offline measurements only has slight deviation, and the overall tendency remains the same. Therefore, the fitting curve can be employed for online position estimation compensations, which can be expressed as

$${\theta }_{comp}=-0.0015i_q^3-0.025i_q+0.018$$

(22)

4. Simulation and Experimental Validation

4.1. Simulation Results

The Matlab/Simulink simulation is developed to verify the time optimized sequence with filterless HF SWVI method. The main parameters of the five-phase IPMSM are shown in

Table 1. Main parameters of IPMSM.

Parameters	Unit	Value
Rated power P	W	2000
Rated speed nr	r/min	1500
Number of phases	/	5
Number of rotor pole-pair	/	9
Stator resistance Rs	Ω	0.5
d-axis inductance Ld	mH	11.8
q-axis inductance Lq	mH	13.7
Moment of inertia J	Kg·m2	0.005
Rated load TL	N.m	11

. The magnitude of injected HF square-wave voltage is 60 V for 2 kHz. The PWM switching frequency is 4 KHz, and the sampling frequency is 8 KHz.

Figure 9 illustrates the effectiveness of the proposed signal processing method without filters. In Figure 9a, the amplitude of the extracted HF current only fluctuates by a few milliamperes in the dynamic process, which means that the proposed method can effectively extract HF currents under dynamic and steady state. In Figure 9b, the comparison of the extracted d-axis FF current with and without the injected HF signals is illustrated. As can be seen, there is a very slight difference between the extracted FF currents under the two conditions, which is caused by the change of zero-crossing time due to the existence of armature winding resistance. It should be noted that the difference can be neglected in dynamic and steady states. Figure 10a shows the comparison results of position estimation error with and without the optimized time sequence. The motor operates with the dynamic step-speed change from 0 to 50 r/min with no load. Compared with the conventional time sequence, the position estimation error is effectively improved with the optimized time sequence. Figure 10b shows the comparison results of steady-state position estimation error, when the motor operates at 300 r/min with no load. It is worth noting that, due to the transition of rotor angle from 2π to 0, there will be a certain spike in the position estimation error. Figure 11 shows the simulation results of the proposed sensorless control with rated load when given speed changes from 0 to 50 rpm at 1.0 s. The maximum position estimation error is 0.26 rad during the start-up region.

4.2. Experimental Results

The sensorless control using the time optimized sequence with HF SWVI method is verified at a 2-kW IPMSM test platform as shown in Figure 12. A DSP-TMS320F28335 is adopted as the MCU to execute the sensorless control algorithm. The magnetic powder brake is coaxially fixed with the IPMSM to produce the load torque. An incremental encoder with 4096 pulses per revolution is used to obtain the actual mechanical position and speed of the IPMSM rotor. The PWM switching frequency of the inverter is 4 KHz, and current sampling frequency is 8 KHz. The magnitude of the injected HF voltage is 60 V for 2 KHz. The PI parameters of d- and q-axis in the current loop are selected as K_p,d = 16.2, K_i,d = 77, K_p,q = 18.8, K_i,q = 66. The PI parameters of speed loop are selected as K_p,n = 0.07, K_i,n = 1.2. The PI parameters of PLL are K_p = 115 and K_i = 3306. The DSDU mode is adopted in the DSP.

Figure 13 shows this comparison under the dynamic state, when the step reference speed changes from 0 to 50 r/min under no-load condition. As can be seen, after the optimized time sequence is adopted, the maximum position estimation error is decreased from 0.204 rad to 0.162 rad, which means that the steady-state error term in (17) has been eliminated effectively. Figure 14 shows the measured actual speed, position estimation error, estimated angle, and actual angle of sensorless control drive with and without optimization under steady state, when the motor operates at 300 r/min with no load. Among these two curves, before optimization, θ_err > 0 indicates that the estimated angle lags behind the actual angle. After optimization, θ_err < 0 indicates that the estimated angle is ahead of the actual angle, and its value fluctuates around −0.042 rad, which is in agreement with the calculation result of (20) and Figure 10b. It is noted that the fluctuation of the angular error mainly comes from the nonlinearity of the inverter. Therefore, the proposed optimized time sequence is effective for improving the accuracy of the position estimation.

To verify the compensation effect of the online look-up table method on the position estimation error caused by the cross-coupling effect, Figure 14 illustrates the measured speed estimation error, q-axis FF current, angle compensation, and position estimation error of sensorless control drive at 150 rpm with step rated load disturbance. It is noted that in order to suppress the influence of digital control on position estimation error, the proposed optimized time sequence is adopted. By comparing Figure 15a,b, after cross-coupling compensation, the position estimation error in the steady state with load is reduced from −0.3 rad to −0.04 rad. Besides, when the load is released, the fluctuation of the position estimation error is reduced from 0.4 rad to 0.28 rad. In Figure 15b, the amount of angle compensation varies smoothly with the q-axis FF current. Therefore, this experiment can demonstrate that the angle compensation data obtained offline under the static state can effectively correct the estimated angle. Figure 16 shows that the sensorless IPMSM operates from 0 to 50 rpm with rated load, and the maximum position estimation error is 0.23 rad during the start-up region.

Figure 17 shows the dynamic-state sensorless control performance at ultra-low speed. The motor operates with the speed of 5 r/min (0.33% of the rated speed) with no load in Figure 17a and rated load in Figure 17b. The absolute value of position estimation error at ultra-low speed with full load condition is about 0.08 rad. From the above experimental results, it can be seen that the sensorless IPMSM drive has a satisfying performance adopting the optimized time sequence.

5. Conclusions

The main objective of this paper is to propose an optimized time sequence for a five-phase IPMSM drive based on the HF SWVI method, which can eliminate the additional position estimation error caused by the digital time delay. Besides, by utilizing the optimized time sequence, the accuracy of position estimation is effectively improved in both steady and dynamic state. Meanwhile, to meet the requirement of high dynamic performance for drive system, the signal processing method without using filters is proposed. Moreover, the online position error compensation is adopted for cross-coupling effect via FEA and offline measurements. All theoretical analyses are verified by experimental and simulation results on a 2-kW IPMSM platform in both the steady and dynamic conditions. The proposed sensorless control scheme in this paper is also applicable to three-phase IPMSM drive.