Permanent magnet synchronous motor (PMSM) is widely used in home appliance applications due to the high power density, high efficiency, and wide speed range. The PMSM includes two major types: the surface-mounted PMSM (SPM), and the interior PMSM (IPM). SPM motors are easier to control due to the linear relationship between the torque and q-axis current. The aim of the field weakening control is to optimize to reach the highest power and efficiency of a PMSM drive. Field weakening control can enable a motor operation over the base speed, expanding the operating limits to reach speeds higher than the rated speed and allow exceptional control across the entire speed and voltage range.

The voltage equations of the mathematical model of an IPMSM can be described in d-q coordinates as shown in Equation 6 and Equation 7.

Figure 3-8 shows the dynamic equivalent circuit of an IPM synchronous motor.

The total electromagnetic torque generated by the IPMSM can be expressed as Equation 8 that the produced torque is composed of two distinct terms. The first term corresponds to the mutual reaction torque occurring between torque current ${\mathrm{i}}_{\mathrm{q}}$ and the permanent magnet ${\mathrm{\psi }}_{\mathrm{m}}$ , while the second term corresponds to the reluctance torque due to the differences in d-axis and q-axis inductance.

In most applications, IPMSM drives have speed and torque constraints, mainly due to inverter or motor rating currents and available DC link voltage limitations respectively. These constraints can be expressed with the mathematical equations Equation 9 and Equation 10.

where

• ${\mathrm{V}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and ${\mathrm{I}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are the maximum allowable voltage and current of the inverter or motor

In a two-level three-phase Voltage Source Inverter (VSI) fed machine, the maximum achievable phase voltage is limited by the DC link voltage and the PWM strategy. The maximum voltage is limited to the value as shown in Equation 11 if Space Vector Modulation (SVPWM) is adopted.

Usually the stator resistance ${\mathrm{R}}_{\mathrm{s}}$ is negligible at high speed operation and the derivative of the currents is zero in steady state, thus Equation 12 is obtained as shown.

The current limitation of Equation 9 produces a circle of radius ${\mathrm{I}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ in the d-q plane, and the voltage limitation of Equation 11 produces an ellipse whose radius ${\mathrm{V}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ decreases as speed increases. The resultant d-q plane current vector must be controlled to obey the current and voltage constraints simultaneously. According to these constraints, three operation regions for the IPMSM can be distinguished as shown in Figure 3-9.

1. Constant Torque Region: MTPA can be implemented in this operation region to provide maximum torque generation.
2. Constant Power Region: Field-weakening control must be employed and the torque capacity is reduced as the current constraint is reached.
3. Constant Voltage Region: In this operation region, deep field-weakening control keeps a constant stator voltage to maximize the torque generation.

In the constant torque region, according to Equation 8, the total torque of an IPMSM includes the electromagnetic torque from the magnet flux linkage and the reluctance torque from the saliency between ${\mathrm{L}}_{\mathrm{d}}$ and ${\mathrm{L}}_{\mathrm{q}}$ . The electromagnetic torque is proportional to the q-axis current ${\mathrm{i}}_{\mathrm{q}}$ , and the reluctance torque is proportional to the multiplication of the d-axis current ${\mathrm{i}}_{\mathrm{d}}$ , the q-axis current ${\mathrm{i}}_{\mathrm{q}}$ , and the difference between ${\mathrm{L}}_{\mathrm{d}}$ and ${\mathrm{L}}_{\mathrm{q}}$ .

Conventional vector control systems of SPM motors only utilizes electromagnetic torque by setting the commanded ${\mathrm{i}}_{\mathrm{d}}$ to zero for non-field-weakening modes. But while the IPMSM utilizes the reluctance torque of the motor, the designer must also control the d-axis current. The aim of the MTPA control is to calculate the reference currents ${\mathrm{i}}_{\mathrm{d}}$ and ${\mathrm{i}}_{\mathrm{q}}$ to maximize the ratio between produced electromagnetic torque and reluctance torque. The relationship between ${\mathrm{i}}_{\mathrm{d}}$ and ${\mathrm{i}}_{\mathrm{q}}$ , and the vectorial sum of the stator current ${\mathrm{I}}_{\mathrm{s}}$ is shown in the following equations.

where

• $\beta$ is the stator current angle in the synchronous (d-q) reference frame

Equation 8 can be expressed as Equation 16 where ${\mathrm{I}}_{\mathrm{s}}$ substituted for ${\mathrm{i}}_{\mathrm{d}}$ and ${\mathrm{i}}_{\mathrm{q}}$ .

Equation 16 shows that motor torque depends on the angle of the stator current vector:

This equation shows the maximum efficiency point can be calculated when the motor torque differential is equal to zero. The MTPA point can be found when this differential, $\frac{\mathrm{d}{\mathrm{T}}_{\mathrm{e}}}{\mathrm{d}\mathrm{\beta }}$ is zero as given in Equation 17.

Following this equation, the current angle of the MTPA control can be derived as in Equation 18.

Thus, the effective d-axis and q-axis reference currents can be expressed by Equation 19 and Equation 20 using the current angle of the MTPA control.

However, as shown in Equation 18, the angle of the MTPA control, ${\beta }_{\mathrm{m}\mathrm{t}\mathrm{p}\mathrm{a}}$ is related to d-axis and q-axis inductance. This means that the variation of inductance impedes the ability to find the exceptional MTPA point. To improve the efficiency of a motor drive, estimate the d-axis and q-axis inductance online, but the parameters ${\mathrm{L}}_{\mathrm{d}}$ and ${\mathrm{L}}_{\mathrm{q}}$ are not easily measured online and are influenced by saturation effects. A robust Look-Up Table (LUT) method provides controllability under electrical parameter variations. Usually, to simplify the mathematical model, the coupling effect between d-axis and q-axis inductance can be neglected. Thus, assume that ${\mathrm{L}}_{\mathrm{d}}$ changes with ${\mathrm{i}}_{\mathrm{d}}$ only, and ${\mathrm{L}}_{\mathrm{q}}$ changes with ${\mathrm{i}}_{\mathrm{q}}$ only. Consequently, d- and q-axis inductance can be modeled as a function of the d-q currents respectively, as shown in Equation 21 and Equation 22.

Reduce the ISR calculation burden by simplifying Equation 18. The motor-parameter-based constant, ${\mathrm{K}}_{\mathrm{m}\mathrm{t}\mathrm{p}\mathrm{a}}$ is expressed instead as Equation 24, where ${\mathrm{K}}_{\mathrm{m}\mathrm{t}\mathrm{p}\mathrm{a}}$ is computed in the background loop using the updated ${\mathrm{L}}_{\mathrm{d}}$ and ${\mathrm{L}}_{\mathrm{q}}$ .

A second intermediate variable, ${\mathrm{G}}_{\mathrm{m}\mathrm{t}\mathrm{p}\mathrm{a}}$ described in Equation 25, is defined to further simplify the calculation. Using ${\mathrm{G}}_{\mathrm{m}\mathrm{t}\mathrm{p}\mathrm{a}}$ , the angle of the MTPA control, ${\beta }_{\mathrm{m}\mathrm{t}\mathrm{p}\mathrm{a}}$ can be calculated as Equation 26. These two calculations are performed in the ISR to achieve a real current angle ${\beta }_{\mathrm{m}\mathrm{t}\mathrm{p}\mathrm{a}}$ .

In all cases, the magnetic flux can be weakened to extend the achievable speed range by acting on the direct axis current i_d. As a consequence of entering this constant power operating region, field weakening control is chosen instead of the MTPA control used in constant power and voltage regions. Since the maximum inverter voltage is limited, PMSM motors cannot operate in such speed regions where the back-electromotive force, almost proportional to the permanent magnet field and motor speed, is higher than the maximum output voltage of the inverter. The direct control of magnet flux is not an option in PM motors. However, the air gap flux can be weakened by the demagnetizing effect due to the d-axis armature reaction by adding a negative i_d. Considering the voltage and current constraints, the armature current and the terminal voltage are limited as Equation 9 and Equation 10. The inverter input voltage (DC-Link voltage) variation limits the maximum output of the motor. Furthermore, the maximum fundamental motor voltage also depends on the PWM method used. In Equation 12, the IPMSM has two factors: one is a permanent magnet value and the other is made by inductance and current of flux.

Figure 3-10 shows the typical control structure is used to implement field weakening. ${\beta }_{\mathrm{f}\mathrm{w}}$ is the output of the field-weakening (FW) PI controller and generates the reference ${\mathrm{i}}_{\mathrm{d}}$ and ${\mathrm{i}}_{\mathrm{q}}$ . Before the voltage magnitude reaches the limit, the input of the PI controller of FW is always positive and therefore the output is always saturated at 0.

The field-weakening control module shown in Figure 3-10 generates the current angle ${\beta }_{\mathrm{f}\mathrm{w}}$ based on input parameters as shown in Figure 3-11.

In a typical application, if using both MTPA and FW control, the switching control module is used to determine angle of application, and then calculate the reference ${\mathrm{i}}_{\mathrm{d}}$ and ${\mathrm{i}}_{\mathrm{q}}$ as shown in Equation 14 and Equation 15. The current angle is chosen as in the following: Equation 27 and Equation 28.