Sensorless Control Based on Search Coils for Six-Phase Permanent Magnet Synchronous Motor

Abstract

Traditional sensorless control has the problem that the accuracy of rotor position estimation depends heavily on the precise motor parameters. In this paper, a sensorless control method based on search coils are proposed for six-phase permanent magnet synchronous motor to eliminate the interference of motor parameters on rotor position estimation. Firstly, the mathematical model of the search coils is described, and the key inductance parameters of the search coils are deduced. Secondly, on the basis of analyzing the induced potential of the search coils, a potential compensation term was introduced to eliminate the potential interference of the transformer under different load conditions, and the rotor position angle was calculated based on the phase-locked loop (PLL). Finally, the effectiveness and feasibility of the control algorithm are verified by simulation and experiment.

1 Introduction

1.1 A Subsection Sample

High power density, high adaptability, high reliability, high precision, low emission and multi-functional composite are the target requirements of motor system performance for the development of energy power and transportation [1]. Multiphase permanent magnet synchronous motor (PMSM) not only has the advantages of high power density, fast dynamic response, high efficiency and large torque inertia ratio, but also can increase the output power of the motor and have the ability of fault-tolerant operation without increasing the power level of the power device. It has broad application prospects in ship electric propulsion, multi-electric/all-electric aircraft and rail transit [2].

Accurate rotor position information is the key to achieve high performance control of PMSM. Position sensors are often installed on the motor shaft to detect the rotor position of the motor in real time, such as photoelectric encoder, resolver and Hall sensor. The use of position sensors not only increases the cost and volume of the system, but also reduces the immunity and reliability of the system. More importantly, some special applications cannot install position sensors, such as shaftless propulsion systems. Therefore, the sensorless control of PMSM is widely used in various industrial occasions [3],

The position sensorless control based on the search coils has attracted attention due to its low cost, strong immunity and independence of motor parameters. The search coil is a group of windings laid in the stator slot of the motor. It was first used to obtain the distribution of the air gap magnetic field [4, 5], and then extended to fault diagnosis [6, 7] and sensorless control [8,9,10]. Classical sensorless control includes high frequency injection method [11], back electromotive force method [12], model reference adaptive observer method [13, 14] and so on. Because the observation accuracy of these algorithms depends heavily on the accuracy of the motor parameters, it is difficult to guarantee the control effect in the case of complex working conditions and high reliability requirements. Since the search coil does not participate in energy conversion, the parameters are not easy to change. Therefore, the position estimation accuracy based on the search coil is not sensitive to the change of motor parameters. Consistent with the classical sensorless control, the sensorless control based on the search coils can calculate the position information of the motor at zero low speed and medium high speed respectively through the salient pole model and fundamental wave model of the motor. The method based on the salient pole model obtains the position signal by injecting high frequency voltage signal into the search coils. Reference [15] injected high-frequency square wave pulse signals into three search coils on the stator slot of the switched reluctance motor, and estimated the rotor position by using the mutual inductance characteristics between the main winding and the search coils generated by the convex polarity of the rotor. The method based on the fundamental wave model is to obtain the rotor position of the motor by solving the air gap magnetic field induced by the search coils. In Reference [16], a direct rotor position calculation method based on the back electromotive force (EMF) of the search coils are proposed for the permanent magnet synchronous motor, which eliminates the error caused by the change of parameters such as stator resistance. At present, the search coils are mainly used for fault analysis, and the research on position sensorless is rarely involved. In view of its application advantages in high reliability occasions, it is worth further study.

In this paper, a position estimation algorithm based on search coils and phase-locked loop is proposed for six-phase PMSM, so as to realize sensorless control. On the basis of expounding the mathematical model of search coils and deducing the key inductance parameters, the compensation term is introduced to eliminate the interference of transformer electromotive force, and the rotor position angle is calculated based on phase-locked loop. Simulation and experimental results verify the feasibility and effectiveness of the control algorithm.

2 Mathematical Model of Six-Phase PMSM

The driving circuit of the six-phase PMSM is shown in Fig. 1.The six-phase winding consists of two sets of three-phase windings with a phase shift of 30°.

Fig. 1

Driving circuit structure of six-phase PMSM

Six-phase PMSM can be equivalent to two three-phase PMSM decoupling control by double dq coordinate transformation, which has a clear physical concept. With the application of the constant amplitude transformation matrix, the mathematical motor model of the ABCXYZ coordinate is transformed into the α₁β₁ and α₂β₂ coordinate systems, and then transformed into the d₁q₁ and d₂q₂ coordinate systems with the application of the rotating coordinate transformation matrix. The relationship between the various coordinate systems is shown in Fig. 2.

In the double dq coordinate system, the voltage equation of the six-phase PMSM can be expressed as:

$$ \left[ \begin{gathered} u_{d1} \hfill \\ u_{q1} \hfill \\ u_{d2} \hfill \\ u_{q2} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {R_{s} } & 0 & 0 & 0 \\ 0 & {R_{s} } & 0 & 0 \\ 0 & 0 & {R_{s} } & 0 \\ 0 & 0 & 0 & {R_{s} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{d1} } \\ {i_{q1} } \\ {i_{d2} } \\ {i_{q2} } \\ \end{array} } \right] + {\text{p}}\left[ {\begin{array}{*{20}c} {\psi_{d1} } \\ {\psi_{q1} } \\ {\psi_{d2} } \\ {\psi_{q2} } \\ \end{array} } \right] + \omega_{e} \left[ {\begin{array}{*{20}c} 0 & { - 1} & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - 1} \\ 0 & 0 & 1 & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\psi_{d1} } \\ {\psi_{q1} } \\ {\psi_{d2} } \\ {\psi_{q2} } \\ \end{array} } \right] $$

(1)

where u_d1,u_q1, u_d2,u_q2 are the DC voltage components in two sets of synchronous rotating coordinate system respectively. ψ_d1, ψ_q1,ψ_d2,ψ_q2 are the corresponding flux components; i_d1,i_q1,i_d2,i_q2 are the corresponding current components; ω_e is the electric angular velocity; p is a differential operator.

The flux linkage equation is expressed as:

$$ \left[ {\begin{array}{*{20}c} {\psi_{d1} } \\ {\psi_{q1} } \\ {\psi_{d2} } \\ {\psi_{q2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {L_{d} } & 0 & {L_{dd} } & 0 \\ 0 & {L_{q} } & 0 & {L_{qq} } \\ {L_{dd} } & 0 & {L_{d} } & 0 \\ 0 & {L_{qq} } & 0 & {L_{q} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{d1} } \\ {i_{q1} } \\ {i_{d2} } \\ {i_{q2} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 1 \\ 0 \\ 1 \\ 0 \\ \end{array} } \right]\psi_{{\text{f}}} $$

(2)

where ψ_f is the flux linkage of permanent magnet; L_d is a direct axis synchronous inductance; L_dd is the mutual inductance between two sets of d-axis windings; L_q is quadrature axis synchronous inductance; L_qq is the mutual inductance between two sets of q-axis windings.

The torque equation is expressed as:

$$ T_{e} = 1.5p_{n} (i_{q1} \psi_{d1} - i_{d1} \psi_{q1} + i_{q2} \psi_{d2} - i_{d2} \psi_{q2} ) $$

(3)

3 Search Coil Principle

3.1 Search Coils Mathematical Model

The search coils proposed in this paper is two windings with orthogonal electrical angles. The rotor position estimation can be realized by its induced potential in the open circuit. The potential is composed of the transformer potential generated by the alternating magnetic field of each phase stator winding and the rotating potential generated by the rotation of the permanent magnet. It can be expressed as:

$$ \left[ {\begin{array}{*{20}c} {u_{{{\text{w1}}}} } \\ {u_{{{\text{w2}}}} } \\ \end{array} } \right] = {\mathbf{M}}_{{{\mathbf{2}} \times {\mathbf{6}}}} {\text{p}}{\mathbf{i}}_{{{\mathbf{6}} \times {\mathbf{1}}}} + \psi_{{\text{f}}} \omega_{e} \left[ {\begin{array}{*{20}c} {k\cos \left( {\theta_{e} + p_{n} \Delta \theta } \right)} \\ {k\cos \left( {\theta_{e} + p_{n} \Delta \theta + \frac{\pi }{2}} \right)} \\ \end{array} } \right] $$

(4)

where u_w1 and u_w2 represent the induced electromotive force of the search coil w1 and w2 respectively; the coefficient k depends on the amplitude of the induced potential of the search coil under no-load condition. ∆θ is the mechanical angle between the search coil w1 and the A-phase winding of the six-phase PMSM, which is determined by the position of the search coils added to the stator.

$$ \begin{gathered} {\mathbf{i}}_{{{\mathbf{6}} \times {\mathbf{1}}}} = \left[ {\begin{array}{*{20}c} {i_{{\text{A}}} } & {i_{{\text{B}}} } & {i_{{\text{C}}} } & {i_{{\text{X}}} } & {i_{{\text{Y}}} } & {i_{{\text{Z}}} } \\ \end{array} } \right]^{{\text{T}}} \hfill \\ {\mathbf{M}}_{{{\mathbf{2}} \times {\mathbf{6}}}} = \left[ {\begin{array}{*{20}c} {M_{{{\text{Aw1}}}} } & {M_{{{\text{Bw1}}}} } & {M_{{{\text{Cw1}}}} } & {M_{{{\text{Xw1}}}} } & {M_{{{\text{Yw1}}}} } & {M_{{{\text{Zw1}}}} } \\ {M_{{{\text{Aw2}}}} } & {M_{{{\text{Bw2}}}} } & {M_{{{\text{Cw2}}}} } & {M_{{{\text{Xw2}}}} } & {M_{{{\text{Yw2}}}} } & {M_{{{\text{Zw2}}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{{\mathbf{1}}} } \\ {{\mathbf{M}}_{{\mathbf{2}}} } \\ \end{array} } \right] \hfill \\ \end{gathered} $$

where M_iwj is the mutual inductance parameter value between the i-phase winding of the stator and the search coil w_j. i represents each phase winding of the stator, and wj represents the search coil w1 and w2. M₁ is the row vector composed of the mutual inductance parameters between the search coil w1 and the stator windings; M₂ is the row vector composed of the mutual inductance parameters between the search coil w2 and the stator windings.

Due to the large difference between the number of turns of the search coil and the number of turns of the stator winding, usually up to dozens of times, the mutual inductance value is not equal to the simple product of the stator self-inductance and the angle cosine value, which needs to be re-derived. At the same time, special attention should be paid to the influence of pitch factor and turns.

The space fundamental magnetic potential F_ag1 in the two-pole motor is:

$$ F_{ag1} = \frac{4}{\pi }\frac{Ni}{2}\cos \theta_{a} $$

(5)

where N is the number of turns of the stator winding; θ_a is the distribution angle with the stator coil magnetic pole axis as the reference zero point.

The corresponding air gap flux density B_ag1 is:

$$ B_{ag1} = \mu_{0} \frac{{F_{ag1} }}{g} = \frac{{2\mu_{0} Ni}}{\pi g}\cos \theta_{a} $$

(6)

where μ₀ is the air permeability; g is the air gap length.

The fundamental air gap flux φ_p per pole is:

$$ \phi_{p} = l\int_{ - \pi /2}^{\pi /2} {B_{ag1} rd\theta_{a} } $$

(7)

where r is the radius of the rotor; l is the axial length of air gap.

The air gap inductance L of the coil is:

$$ L = \frac{\psi }{i} = \frac{{N\phi_{p} }}{i} = \frac{{4\mu_{0} N^{2} lr}}{\pi g} $$

(8)

For the multi-pole distribution winding with the pole pair of p_n and the number of turns in series of N₁ and the winding coefficient of k_w1, N is replaced by the effective number of turns per pole (k_w1N₁/p_n) in Formula (8), and the air gap inductance can be obtained as:

$$ L = \frac{{4\mu_{0} lr}}{\pi g}\left( {\frac{{k_{w1} N_{1} }}{{{\text{p}}_{{\text{n}}} }}} \right)^{2} $$

(9)

Therefore, for the two sets of windings with winding distribution factors of k_w1 and k_w2 and turns of N₁ and N₂ respectively in the same motor when the magnetic pole axis insulation angle is α, the mutual inductance of the two sets of windings is:

$$ M_{12} = \frac{{4\mu_{0} }}{\pi }\left( {\frac{{k_{w1} N_{1} }}{{{\text{p}}_{{\text{n}}} }}} \right)\left( {\frac{{k_{w2} N_{2} }}{{{\text{p}}_{{\text{n}}} }}} \right)\frac{lr}{g}\cos \alpha $$

(10)

The general ratio between mutual inductance M₁₂ and air gap inductance L is:

$$ \frac{{M_{12} }}{L} = \frac{{k_{{{\text{w2}}}} N_{2} }}{{k_{{{\text{w1}}}} N_{1} }}\cos \alpha $$

(11)

According to Eq. (11), the mutual inductance parameter value between the stator winding and the search coil can be calculated by the air gap inductance of each stator winding.

3.2 Rotor Position Estimation Based on Search Coils

The induced potential of the search coils consists of two parts, which are the transformer potential of the first term and the moving potential of the second term in Eq. (4). The information of the rotor position estimation only exists in the moving potential, and as the load increases, the amplitude of the transformer potential will also increase, and the proportion of the moving potential will decrease, so the interference to the rotor position estimation will also increase. Therefore, it is necessary to compensate the induced potential of the search coils to eliminate the transformer potential.

$$ \left[ {\begin{array}{*{20}c} {u_{{{\text{c\_T1}}}} } \\ {u_{{{\text{s\_T1}}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {u_{{{\text{w1}}}} } \\ {u_{{{\text{w2}}}} } \\ \end{array} } \right] - {\mathbf{M}}_{{{\mathbf{2}} \times {\mathbf{6}}}} {\text{p}}{\mathbf{i}}_{{{\mathbf{6}} \times {\mathbf{1}}}} = \psi_{{\text{f}}} \omega_{{\text{e}}} \left[ {\begin{array}{*{20}c} {k\cos \left( {\theta_{e} + p_{n} \Delta \theta } \right)} \\ {k\cos \left( {\theta_{e} + p_{n} \Delta \theta + \frac{\pi }{2}} \right)} \\ \end{array} } \right] $$

(12)

The rotor position angle can be obtained by inputting the induced potential after potential compensation into the phase-locked loop. In the induced electromotive force, the sine and cosine functions of $\hat{\theta }_{e}$ are multiplied by the cosine and sine of the estimated value of the rotor position, and the error ε between the actual value and the estimated value is:

$$ \varepsilon = \sin \theta_{e} \cos \hat{\theta }_{e} - \cos \theta_{e} \sin \hat{\theta }_{e} = \sin (\theta_{e} - \hat{\theta }_{e} ) $$

(13)

Finally, ε can output the estimated rotor position through the PI regulator. The rotor position estimation control block diagram based on the search coils is shown in Fig. 3.

Fig. 2

Double dq coordinate transformation of six-phase PMSM

The six-phase PMSM sensorless vector control block diagram based on the search coils are shown in Fig. 4. The rotor position estimation and speed estimation are obtained by processing the search coils voltage u and the motor phase current i obtained by sampling. The rotor position estimation is used to realize the coordinate transformation, which is used to realize the double closed-loop vector control of the motor speed and current based on $i_{{{\text{d}}1}}^{*} = 0$, $i_{{{\text{d}}2}}^{*} = 0$.

Fig. 3

Control block diagram of rotor position estimation based on search coils

4 Simulation and Experimental Verification

In order to verify the effectiveness and feasibility of the proposed control strategy, a model is built for simulation verification, and compared with the experimental results of the actual six-phase permanent magnet synchronous motor. The six-phase PMSM parameters used are shown in Table 1.

Table 1 Parameters of six-phase PMSM

4.1 Simulation of No-Load Condition

In order to verify the feasibility and effectiveness of the control strategy under no-load, the simulation experiment is carried out under the condition of no-load and speed of 300 r/min. The steady-state simulation and experimental results are shown in Fig. 5. In order to verify the dynamic performance, the speed dynamic acceleration simulation is carried out. The speed gradually increases from 100 r/min to 300 r/min at 13.5 s, and gradually decreases from 300 r/min to 100 r/min at 108 s. The simulation results are shown in Fig. 6.

Fig. 4

PMSM sensorless vector control block diagram

Fig. 5

Simulation and experiment results in steady state process

Fig. 6

Simulation and experiment results of dynamic speed up and down process

4.2 Comparative Analysis

The position tracking error based on the search coils is further extracted from the experimental waveform. As shown in Table 2, the position error is 0.15 rad. For the static and dynamic position tracking error performance of multi-phase permanent magnet motor sensorless control, Table 2 shows that the maximum steady-state position error is 2.39%, the steady-state speed error is less than 1 rpm, and the dynamic speed error is less than 2 rpm, which more effectively verifies the feasibility of rotor position estimation of permanent magnet motor based on search coils.

Table 2 Position tracking error analysis based on search coils

5 Conclusion

The sensorless control of six-phase PMSM can be realized based on the search coils. And the rotor position obtained based on the search coils is independent of parameters such as stator resistance. Therefore, the temperature rise of the motor and the change of the environment will not affect the accuracy of position estimation. This provides a new idea for rotor position estimation of multiphase permanent magnet motors applied to shaftless propulsion systems.