TIDUF67 Design guide

TIDUF67 April 2024 – December 2024

3.1.3.1.2 Rotor Position and Speed Estimation with PLL

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 Equation 18 and Equation 19 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 Equation 28, Equation 29, and Equation 30

Equation 28.

e_{α} = Ecos (θ_{e})

Equation 29.

e_{β} = Esin (θ_{e})

Equation 30.

E = ω_{e} λ_{pm}

the position error can be defined as:

Equation 31.

ε = {\hat{e}}_{β} \cos ({\hat{θ}}_{e}) - {\hat{e}}_{α} \sin ({\hat{θ}}_{e}) = Esin (θ_{e}) \cos ({\hat{θ}}_{e}) - Ecos (θ_{e}) \sin ({\hat{θ}}_{e}) = Esin (θ_{e} - {\hat{θ}}_{e})

Where E is the magnitude of the EEMF, which is proportional to the motor speed ω_e. When Equation 32, the Equation 31 can be simplified as

Equation 32.

(θ_{e} - {\hat{θ}}_{e}) < \frac{π}{2}

Equation 33.

ε = E (θ_{e} - {\hat{θ}}_{e})

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

Equation 34.

ε_{n} = θ_{e} - {\hat{θ}}_{e}

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:

Equation 35.

\frac{{\hat{θ}}_{e}}{θ_{e}} = \frac{k_{p} s + k_{i}}{s^{2} + k_{p} s + k_{i}} = \frac{2 {ξω}_{n} s + ω_{n}^{2}}{s^{2} + 2 {ξω}_{n} s + ω_{n}^{2}}

where the k_p and k_i are the proportional and the integral gains of the standard PI regulator, the natural frequency ω_n and the damping ratio ξ is given as

Equation 36.

k_{p} = 2 {ξω}_{n}, k_{i} = ω_{n}^{2}

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