With the arc tangent method, the accuracy of the position and velocity estimations are affected due to the existence of noise and harmonic components. To eliminate this issue, the PLL model can be used for velocity and position estimations in the sensorless control structure of the IPMSM. The PLL structure used with SMO is illustrated in Section 3.1.3.1.1. The back-EMF estimations ${\widehat{e}}_{\alpha }$ and ${\widehat{e}}_{\beta }$ can be used with a PLL model to estimate the motor angular velocity and position as shown in Figure 3-10.

Figure 3-10 Block Diagram of Phase Locked Loop Position Tracker

Since $e_{\alpha }=E\cos \left({\theta }_e\right)$ , $e_{\beta }=E\sin \left({\theta }_e\right)$ and $E={\omega }_e{\lambda }_{pm}$ , the position error can be defined as:

Where E is the magnitude of the EEMF, which is proportional to the motor speed ${\omega }_e$ . When $({\theta }_e-{\hat{\theta }}_e)<\frac{\mathrm{\pi }}{2}$ , the Equation 20 can be simplified as

Further, the position error after the normalization of the EEMF can be obtained:

According to the analysis, the simplified block diagram of the quadrature phaselocked loop position tracker can be obtained as shown in Figure 3-11. The closed-loop transfer functions of the PLL can be expressed as:

where the $k_p$ and $k_i$ are the proportional and the integral gains of the standard PI regulator, the natural frequency ${\omega }_n$ and the damping ratio $\xi$ is given:

Figure 3-11 Simplified Block Diagram of Phase Locked Loop Position Tracker