# Self-Sensing Six-Phase PMSM Drive Based on Back-EMF Measurement
## 6 Operation Mode with the Faulty Speed/Position Sensor  
The emergency three-phase operation is shown in this part of the paper. A resolver fault is assumed. The control procedure is the same as the three-phase operation mode described in the previous section, except for the speed/position feedback, which is now obtained from back-EMF voltages measured on the winding of the second system D-E-F. The results for sensorless control are presented in Figures 15-20. The value of the back EMF is proportional to the rotor speed, so the proposed method does not work in the low-speed region. This technique is based on the phase angle of measured voltages, and the position of the measured three-phase signal is unrecognizable in the noise if the amplitude is small. The results of the acceleration from 300 rpm to 3000 rpm are shown in Figures 15-17. Data from the resolver are shown in Figures 17, 19 and 20 only to verify the correctness of the estimated speed/position, and they are not used in the control algorithm. The startup shown in Figure 17 has been executed for no-load conditions. Figure 15: Phase current during the startup transient of the emergency three-phase operation. Figure 16: Current in transformed subsystems during the startup transient of the emergency three-phase operation: (**a**) \(\alpha\)–\(\beta\) subsystems; (**b**) \(z1\)–\(z2\) subsystem. Figure 17: Startup transient of the emergency three-phase operation: (**a**) rotor speed; (**b**) current in \(d\)–\(q\) subsystem. Figure 18: Vector diagram of the second three-phase system D–E–F in the synchronous reference frame: (**a**) six-phase operation; (**b**) three-phase operation. As follows from the mathematical model (1)-(4), the increase in the stator current in the active winding generates an induced voltage in the disconnected winding. This state creates a phenomenon called the armature reaction. The position of the measured voltage represented by the space vector \(E_{s2}\) is displaced from the actual rotor flux position by the load angle \(\vartheta_{L}\), as shown in Figure 18[19]. The armature reaction is compensated, as described in (19) and (20), where the flux-producing component of the stator current is considered to be equal to zero. \[\vartheta_{L}=\tan^{-1}\biggl{(}\frac{M_{q12}\;i_{q1}}{M_{d12}\;i_{q1}+\psi_{ PM}}\biggr{)} \tag{19}\]  
\[\vartheta_{est.\ compensated}=\vartheta_{est.}-\vartheta_{L} \tag{20}\]  
The position behavior corresponding to the no-load startup (Figure 17) is shown for the transient at approximately 2000 rpm in Figure 19a. The steady-state operation at 3000 rpm is presented in Figure 19b. The next test was performed at the steady-state 3000 rpm operation, where the mechanical load was increased by the DC machine with 0.5 kW power coupled mechanically to the tested six-phase PMSM. The load torque is at the level corresponding to 9 A of the stator current in the three-phase operation of the six-phase PMSM. The compensation of the position error due to the mutual coupling between two sets of three-phase windings is based on the mutual inductances and the voltage constant, as defined in (19). The position correction can be affected by a change in parameters. Figure 19: Electrical rotor position: (**a**) startup transient of the emergency three-phase operation at approximately 2000 rpm; (**b**) no-load steady state of the emergency three-phase operation at 3000 rpm. Figure 20: Electrical rotor position: (**a**) load steady state of the emergency three-phase operation; (**b**) zoom of the load steady state at the emergency three-phase operation. However, the portion of the q-axis mutual cross-coupling is much lower than what is sketched in Figure 18. The real value of the position displacement is shown in Figure 20. The compensation is used only as the position corrector after the estimation process, so the speed is not directly influenced by the parameters, as is standard for many model-based back-EMF observers.