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. Section 2.4.2.2.1.2 illustrates the PLL structure used with SMO. The back-EMF estimations ${\widehat{\mathrm{e}}}_{\alpha }$ and ${\widehat{\mathrm{e}}}_{\beta }$ can be used with a PLL model to estimate the motor angular velocity and position as shown in Figure 2-22.

Since ${\mathrm{e}}_{\alpha }=\mathrm{E}\cos \left({\mathrm{\theta }}_{\mathrm{e}}\right)$ , ${\mathrm{e}}_{\beta }=\mathrm{E}\sin \left({\mathrm{\theta }}_{\mathrm{e}}\right)$ , and $\mathrm{E}={\omega }_{\mathrm{e}}{\lambda }_{\mathrm{p}\mathrm{m}}$ , the position error can be defined as Equation 24:

where

• E is the magnitude of the EEMF, which is proportional to the motor speed ${\omega }_{\mathrm{e}}$

When $({\mathrm{\theta }}_{\mathrm{e}}-{\hat{\mathrm{\theta }}}_{\mathrm{e}})<\frac{\mathrm{\pi }}{2}$ , then Equation 24 can be simplified as Equation 25.

Further, the position error after the normalization of the EEMF can be obtained (Equation 26):

According to the analysis, the simplified block diagram of the quadrature phase-locked loop position tracker can be obtained as shown in Figure 2-23. The closed-loop transfer functions of the PLL can be expressed as Equation 27:

where

• ${\mathrm{k}}_{\mathrm{p}}$ and ${\mathrm{k}}_{\mathrm{i}}$ are the proportional and the integral gains of the standard PI regulator

The natural frequency ${\omega }_{\mathrm{n}}$ and the damping ratio $\xi$ are given in Equation 28: