2 PLL Type Estimator

The estimator used in this application note is an adaptation of the one presented in "AN1162 - Sensorless Field Oriented Control (FOC) of an AC Induction Motor (ACIM)” (see References), but applied to PMSM motor particularities.

The estimator has PLL structure. Its operating principle is based on the fact that the d-component of the Back Electromotive Force (BEMF) must be equal to zero at a steady state functioning mode. The block diagram of the estimator is shown in Figure 2-1.

Starting from the closed loop shown in Figure 2-1, the estimated speed (ω _Restim) of the rotor is integrated to obtain the estimated angle, as shown in Equation 2-1:

The estimated speed, ω _Restim, is obtained by dividing the q-component of the BEMF value with the voltage constant, K_Φ, as shown in Equation 2-2.

Considering the initial estimation premise (the d-axis value of BEMF is zero at steady state) shown in Equation 2-2, the BEMF q-axis value, E_qf, is corrected using the d-axis BEMF value, E_df, depending on its sign. The BEMF d-q component’s values are filtered with a first order filter, after their calculation with the Park transform, as indicated in Equation 2-3.

With the fixed stator frame, Equation 2-4 represents the stators circuit equations.

In Equation 2-4, the terms containing α – β were obtained from the three-phase system’s corresponding measurements through Clarke transform. L_S and R_S represent the per phase stator inductance and resistance, respectively, considering Y (star) connected stator phases. If the motor is Δ (delta) connected, the equivalent Y connection phase resistance and inductance should be calculated and used in the equations above.

Figure 2-2 denotes the estimator’s reference electrical circuit model. The A, B and C terminals of the motor are connected to the inverter’s output terminals. The voltages, V_A, V_B and V_C, represent the phase voltages applied to the motor’s stator windings. V_AB, V_BC and V_CA, represent the line voltages between the inverter’s legs, while the phase currents are I_A, I_B and I_C.

Taking one step forward concerning the equations implementation in the control system, the voltages V_α and V_β, implied in estimator’s Equation 2-4 are a previous cycle calculation of the FOC, being fed to the Space Vector Modulation (SVM) block on the previous step of control, but also to the estimator block current step. I_α and I_β are Clarke transform results from the phase currents, which are read every estimator cycle.

The resulting E_α and E_β values of BEMF are translated to the rotating reference frame of the rotor flux through the Park transform resulting in E_d and E_q values, which conform to Equation 2-3. The angle ρ_estim, used in Park transformation is calculated on the previous execution cycle of the estimator. The d-q values of BEMF are then filtered using first order filters, entering the main condition of the estimator, based on E_d being equal to ‘0’.

Equation 2-2 reflects the calculation of ω _Restim, which is the resulting electrical speed. The integrated electrical speed provides the angle (ρ_estim) between the rotor flux and the α – β fixed stator frame. In Equation 2-2, K_Φ denotes the voltage constant as indicated in Table 3-1. The K_Φ used in the electrical speed computation, is shown in Equation 2-5.

MotorEstimParm.qInvKFi represents: Where: P = Number of pole pairs and the other inputs indicated previously

The speed feedback is filtered using a first order filter identical with the one used in the BEMF case. The filter’s generic form is shown in Equation 2-6:

Where: y(n) = Current cycle filter output y(n – 1) = Previous cycle filter output x(n) = Current cycle filter input Kfilter = Filter constant

The DC type values at the filter’s output should be free of noise from the ADC acquisition or high-frequency variations introduced by the software calculations. The filter’s tuning depends on how fast the filtered values (BEMF d-q components and electrical speed) can vary, allowing for sufficient bandwidth, which reduces the possibility of useful signal loss. In the case of BEMF dq components, two situations can be identified:

• High speed. In the Flux weakening mode, where their variation is slow due to the lack of sudden torque change or high acceleration ramp.
• Low speed.

The speed variation depends on the mechanical constant of the motor (and the load coupled on the motor’s shaft) and the slope of the ramp up or ramp down limits on the speed reference, whichever is faster.