TIDUF67 Design guide

TIDUF67 April 2024 – December 2024

3.1.5.1 Field Weakening (FW) and Maximum Torque Per Ampere (MTPA) Control

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. However, the IPMSM has electromagnetic and reluctance torques due to a large saliency ratio. The total torque is non-linear with respect to the rotor angle. As a result, the MTPA technique can be used for IPM motors to optimize torque generation in the constant torque region. 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 rated speed and allow optimal 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 37 and Equation 38.

Equation 37.

v_{d} = L_{d} \frac{{di}_{d}}{dt} + {R_{s} i}_{d} - {pω}_{m} L_{q} i_{q}

Equation 38.

v_{q} = L_{q} \frac{{di}_{q}}{dt} + {R_{s} i}_{q} + {pω}_{m} L_{d} i_{d} + {pω}_{m} ψ_{m}

The dynamic equivalent circuit of an IPM synchronous motor is shown in Figure 3-12.

TIDM-02018 Equivalent Circuit of an IPM Synchronous Motor

Figure 3-12 Equivalent Circuit of an IPM Synchronous Motor

The total electromagnetic torque generated by the IPMSM can be expressed as Equation 40 that the produced torque is composed of two distinct terms. The first term corresponds to the mutual reaction torque occurring between torque current i_q and the permanent magnet,

Equation 39.

ψ_{m}

while the second term corresponds to the reluctance torque due to the differences in d-axis and q-axis inductance.

Equation 40.

T_{e} = \frac{3}{2} p [{ψ_{m} i_{q} + (L_{d} - L}_{q}) i_{d} i_{q}]

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 41 and Equation 42.

Equation 41.

I_{a} = \sqrt{i_{d}^{2} + i_{q}^{2}} \leq I_{\max}

Equation 42.

V_{a} = \sqrt{v_{d}^{2} + v_{q}^{2}} \leq V_{\max}

Where V_maxand I_max 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 43 if Space Vector Modulation (SVPWM) is adopted.

Equation 43.

\sqrt{v_{d}^{2} + v_{q}^{2}} \leq v_{\max} = \frac{v_{dc}}{\sqrt{3}}

Usually the stator resistance R_s is negligible at high speed operation and the derivative of the currents is zero in steady state, thus Equation 44 is obtained as shown.

Equation 44.

\sqrt{L_{d}^{2} (i_{d} + \frac{ψ_{pm}}{L_{d}})^{2} + L_{q}^{2} i_{q}^{2}} \leq \frac{V_{\max}}{ω_{m}}

The current limitation of Equation 41 produces a circle of radius I_max in the d-q plane, and the voltage limitation of Equation 43 produces an ellipse whose radius V_max 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 3-13.

TIDM-02018 IPMSM Control Operation Regions

Figure 3-13 IPMSM Control Operation Regions

Constant Torque Region: MTPA can be implemented in this operation region to maintain maximum torque generation.
Constant Power Region: Field weakening control must be employed and the torque capacity is reduced as the current constraint is reached.
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 40, the total torque of an IPMSM includes the electromagnetic torque from the magnet flux linkage and the reluctance torque from the saliency between L_d and L_q. The electromagnetic torque is proportional to the q-axis current i_q, and the reluctance torque is proportional to the multiplication of the d-axis current i_d, the q-axis current i_q, and the difference between L_d and L_q.

Conventional vector control systems of a SPM motors only utilizes electromagnetic torque by setting the commanded i_d to zero for non-field weakening modes. But an IPMSM utilizes the reluctance torque of the motor, d-axis current must be controlled as well. The aim of the MTPA control is to calculate the reference currents i_d and i_q to maximize the ratio between produced electromagnetic torque and reluctance torque. The relationship between i_d and i_q, and the vectorial sum of the stator current I_s is shown in the following equations.

Equation 45.

I_{s} = \sqrt{i_{d}^{2} + i_{q}^{2}}

Equation 46.

I_{d} = I_{s} \cos β

Equation 47.

I_{q} = I_{s} \sin β

Where β is the stator current angle in the synchronous (d-q) reference frame. Equation 40 can be expressed as Equation 48 where I_s substituted for i_dand i_q.

Equation 48 shows that motor torque depends on the angle of the stator current vector; as such

Equation 48.

T_{e} = \frac{3}{2} {pI}_{s} \sin β [{ψ_{m} + (L_{d} - L}_{q}) I_{s} \cos β]

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

Equation 49.

\frac{d T_{e}}{d β}

is zero as given in Equation 50.

Equation 50.

\frac{{dT}_{e}}{dβ} = \frac{3}{2} p [ψ_{m} I_{s} \cos β {+ (L_{d} - L}_{q}) I_{s}^{2} \cos 2 β] = 0

Following, the current angle of the MTPA control can be derived as in Equation 51.

Equation 51.

β_{mtpa} = \cos^{- 1} \frac{- ψ_{m} + \sqrt{ψ_{m}^{2} + {8 \times (L_{d} - L_{q})}^{2} \times I_{s}^{2}}}{4 \times (L_{d} - L_{q}) \times I_{s}}

Thus, the effective d-axis and q-axis reference currents can be expressed by Equation 52 and Equation 53 using the current angle of the MTPA control.

Equation 52.

I_{d} = I_{s} \times \cos β_{mtpa}

Equation 53.

I_{q} = I_{s} \times \sin β_{mtpa}

However, as shown in Equation 51, the angle of the MTPA control, β_mtpa is related to d-axis and q-axis inductance. This means that the variation of inductance impedes the ability to find the best MTPA point. To improve the efficiency of a motor drive, the d-axis and q-axis inductance must be estimated online, but the parameters L_d and L_q are not easily measured online and are influenced by saturation effects. A robust Look-Up Table (LUT) method maintains 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, assumes that L_d changes with i_d only, and L_q changes with i_q only. Consequently, d- and q-axis inductance can be modeled as a function of the d-q currents respectively, as shown in Equation 54 and Equation 55.

Equation 54.

L_{d} = f_{1} (i_{d}, i_{q}) = f_{1} (i_{d})

Equation 55.

L_{q} = f_{2} (i_{q} {, i}_{d}) = f_{2} (i_{q})

To reduce the ISR calculation burden by simplifying Equation 51. The motor-parameter-based constant, K_mtpa is expressed instead as Equation 56, where K_mtpa is computed in the background loop using the updated L_d and L_q.

Equation 56.

K_{mtpa} = \frac{ψ_{m}}{4 \times (L_{q} - L_{d})} = 0.25 \times \frac{ψ_{m}}{(L_{q} - L_{d})}

Equation 57.

β_{mtpa} = \cos^{- 1} (K_{mtpa} \div I_{s} - \sqrt{{(K_{mtpa} \div I_{s})}^{2} + 0.5})

A second intermediate variable, G_mtpa described in Equation 58, is defined to further simplify the calculation. Using G_mtpa, the angle of the MTPA control, β_mtpacan be calculated as Equation 59. These two calculations are performed in the ISR to achieve a real current angle β_mtpa.

Equation 58.

G_{mtpa} = K_{mtpa} \div I_{s}

Equation 59.

β_{mtpa} = \cos^{- 1} (G_{mtpa} - \sqrt{G_{mtpa}^{2} + 0.5})

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 41 and Equation 42. 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 44, the IPMSM has two factors: one is a permanent magnet value and the other is made by inductance and current of flux.

Figure 3-14 shows the typical control structure is used to implement field weakening.

β_fw is the output of the field weakening (FW) PI controller and generates the reference i_d and i_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.

TIDM-02018 Block Diagram of Field-Weakening and Maximum Torque per Ampere Control

Figure 3-14 Block Diagram of Field-Weakening and Maximum Torque per Ampere Control

There are two control modules in the motor drive FOC system: one is MTPA control and the other one is field weakening control. These two modules generate current angle β_mtpa and β_fw respectively based on input parameters as show in Figure 3-15.

TIDM-02018 Current Phasor Diagram of an IPMSM During FW and MTPA

Figure 3-15 Current Phasor Diagram of an IPMSM During FW and MTPA

The switching control module is used to decide which angle can be applied, and then calculate the reference i_d and i_q as shown in Equation 46 and Equation 47. The current angle is chosen as following Equation 60 and Equation 61.

Equation 60.

β = β_{fw} if β_{fw} > β_{mtpa}

Equation 61.

β = β_{mpta} if β_{fw} < β_{mtpa}

Figure 3-16 shows the overall block diagram of sensorless FOC of PMSM using eSMO with field weakening control (FWC) and maximum torque per ampere (MTPA) in this reference design.

TIDM-02018 Sensorless FOC of PMSM using eSMO with FWC and MTPA

Figure 3-16 Sensorless FOC of PMSM using eSMO with FWC and MTPA