4 Equation-based Flux Weakening

The back-EMF of the motor increases linearly with speed. Therefore, to counter the back-EMF and support the load, the applied voltage increases linearly with speed. The maximum DC bus voltage (V_BUS) is limited by practical considerations, such as winding insulation, motor safe operating zone. In a SVPWM modulation scheme, the maximum applicable phase voltage is shown in Equation 4-1. For a given PMSM motor and DC bus voltage, the maximum applicable phase voltage is not sufficient to counter the generated back-EMF voltage and support the load beyond the base speed of the motor. To increase the speed of the motor beyond base speed, the generated back-EMF must be reduced, which is achieved by weakening the rotor magnetic flux, and is termed Flux weakening mode.

Equation 4-2 shows the steady state voltage equation of the D-Q axis of the PMSM motor. The q-axis applied voltage, V_qs counteracts the back EMF (ω · Ψ_PM) to drive the load at a particular speed. V_qs voltage is limited by maximum DC bus voltage. In order to increase the speed of the motor without increasing the DC bus voltage, the back-EMF must be reduced by lowering the rotor magnet flux linkage, Ψ_PM. However, in a PMSM motor the flux linkage due to the rotor permanent magnets (Ψ_PM) is constant. Thus, in PMSM motor, flux weakening is achieved by injecting negative current in d-axis (i.e., -i_ds such that it counteracts the rotor magnet flux linkage, Ψ_PM).

During flux weakening, the negative d-axis current must be injected such that the applied voltage magnitude always lies on the voltage limit circle, V_max, as shown in Figure 1-4. Therefore, the d-axis current, i_ds can be calculated, as shown in Equation 4-3.

The maximum value of q-axis current is limited during flux weakening in order to ensure that the maximum magnitude of the injected current does not exceed the maximum rated motor current, as shown in Equation 4-4. Therefore, operation in flux weakening causes reduction in torque generating capability.

A concern during the Flux Weakening mode is voltage limitation of the inverter. This voltage limitation is translated to the maximum achievable values for d-q current components. If both components would have followed their reference values, their resulting scalar summation value would overlap the maximum value of ‘1’. Therefore, the maximum current permitted for q-component of the current (the torque component of the current) will result from prioritizing the d-component of the current responsible flux weakening, which is more important due to air gap flux weakening purposes. Figure 4-1 presents this dynamic adjustment translated to the d-q component of the voltages (d-component of voltage prioritizing).

Because estimator performance depends drastically on the parameters of the motor, the experimental results keep this premise for the conditions of the measurements. The first dependence of rotor resistance and flux constant of the motor is the temperature. A high torque is obtained using the maximum current input, resulting in high Joule losses, resulting in an increased motor temperature. This has a negative effect on the validity of the estimator’s output. The intention of this application note is not to correct or compensate for the effect of temperature on the estimation. The compensation of parameters with temperature is possible, but it varies considerably from one motor type to another, the working conditions and functional mode. As a consequence, the test results indicated below have a premise that limits the temperature effect on the estimator’s output. The time for the achieved torque is limited to one minute of continuous functioning at room temperature, see Table 4-1.

It may be observed that the phase current measured for the last two entries in Table 4-1, corresponding to the flux weakening operation, are higher than the ones immediately preceding them, in normal operation speed.