MTPA Control Reference

Compute reference currents for maximum torque per ampere (MTPA) and field-weakening operation

Since R2020a

• 

Libraries:
Motor Control Blockset / Controls / Control Reference

Description

The MTPA Control Reference block computes the d-axis and q-axis reference current values for maximum torque per ampere (MTPA) and field-weakening operations. The computed reference current values result in efficient output for the permanent magnet synchronous motor (PMSM).

You can specify the reference torque and feedback mechanical speed and the block outputs the corresponding d- and q-axes reference current values for MTPA and field-weakening operations. For a given mechanical speed, in addition to the highest possible torque values, the block also supports reference torque values that are less than the rated torque of a PMSM.

The block computes the reference current values by solving mathematical relationships.

In addition, you can use the Vdc input method parameter to configure the block to accept a fixed reference DC voltage through the DC voltage (V) parameter or a variable reference DC voltage through a separate input port V_dc.

These equations describe the computation of reference d-axis and q-axis current values by the block:

Mathematical Model of PMSM

These model equations describe the dynamics of the PMSM in the rotor flux reference frame:

$v_d=i_d{R}_s+ \frac{d{\lambda }_d}{dt} - {\omega }_e{L}_q{i}_q$

$v_q=i_q{R}_s+ \frac{d{\lambda }_q}{dt} + {\omega }_e{L}_d{i}_d+ {\omega }_e{\lambda }_{pm}$

${\lambda }_d=L_d{i}_d+ {\lambda }_{pm}$

${\lambda }_q=L_q{i}_q$

$T_e= \frac{3}{2}p\left({\lambda }_{pm}{i}_q+\left(L_d- L_q\right)i_d{i}_q\right)$

$T_e- T_L=J\frac{d{\omega }_m}{dt} +B{\omega }_m$

where:

• $v_d$ is the d-axis voltage (volts).

• $v_q$ is the q-axis voltage (volts).

• $i_d$ is the d-axis current (amperes).

• $i_q$ is the q-axis current (amperes).

• $R_s$ is the stator phase winding resistance (ohms).

• ${\lambda }_{pm}$ is the permanent magnet flux linkage (weber).

• ${\lambda }_d$ is the d-axis flux linkage (weber).

• ${\lambda }_q$ is the q-axis flux linkage (weber).

• ${\omega }_e$ is the electrical speed corresponding to frequency of stator voltages (radians/ sec).

• ${\omega }_m$ is the rotor mechanical speed (radians/ sec).

• $L_d$ is the d-axis winding inductance (henry).

• $L_q$ is the q-axis winding inductance (henry).

• $T_e$ is the electromechanical torque produced by the PMSM (Nm).

• $T_L$ is the load torque (Nm).

• $$p$$ is the number of motor pole pairs.

• $$J$$ is the inertia coefficient (kg-m²).

• $$B$$ is the friction coefficient (kg-m²/ sec).

Base Speed

Base speed is the maximum motor speed at the rated voltage and rated load, outside the field-weakening region. These equations describe the computation of the motor base speed.

The inverter voltage constraint is defined by computing the d-axis and q-axis voltages:

$v_{do}=- {\omega }_e{L}_q{i}_q$

$v_{qo}={\omega }_e\left(L_d{i}_d+ {\lambda }_{pm}\right)$

$v_{max}= \frac{v_{dc} }{\sqrt{3}} -R_s{i}_{max} \ge \sqrt{v_{do}^2+ v_{qo}^2}$

The current limit circle defines the current constraint which can be considered as:

$i_{max}^2= i_{d }^2+ i_q^2$

In the preceding equation, $i_d$ is zero for surface PMSMs. For interior PMSMs, the block considers values of $i_d$ and $i_q$ corresponding to MTPA.

Using these equations, we can compute the base speed as:

$${\omega }_{base}= \frac{1}{p}\cdot \frac{v_{max}}{\sqrt{{\left(L_q{i}_q\right)}^2+{\left(L_d{i}_d+ {\lambda }_{pm}\right)}^2}}$$

where:

• ${\omega }_e$ is the electrical speed corresponding to frequency of stator voltages (radians/ sec).

• ${\omega }_{base}$ is the mechanical base speed of the motor (radians/ sec).

• $i_d$ is the d-axis current (amperes).

• $i_q$ is the q-axis current (amperes).

• $v_{do}$ is the d-axis voltage when $i_d$ is zero (volts).

• $v_{qo}$ is the q-axis voltage when $i_q$ is zero (volts).

• $L_d$ is the d-axis winding inductance (henry).

• $L_q$ is the q-axis winding inductance (henry).

• $R_s$ is the stator phase winding resistance (ohms).

• ${\lambda }_{pm}$ is the permanent magnet flux linkage (weber).

• $v_d$ is the d-axis voltage (volts).

• $v_q$ is the q-axis voltage (volts).

• $v_{max}$ is the maximum fundamental line to neutral voltage (peak) supplied to the motor (volts).

• $v_{dc}$ is the dc voltage supplied to the inverter (volts).

• $i_{max}$ is the maximum phase current (peak) of the motor (amperes).

• $$p$$ is the number of motor pole pairs.

Surface PMSM

For a surface PMSM, you can achieve maximum torque by using zero d-axis current when the motor is below the base speed. For the field-weakening operation, the block calculates the reference d-axis current using the constant-voltage-constant-power control (CVCP) algorithm defined by these equations:

If ${\omega }_m\le {\omega }_{base}$:

• $i_{d\_mtpa}= 0$

• $$i_{q\_mtpa}= \frac{T^{ref}}{\frac{3}{2}\cdot p\cdot {\lambda }_{pm}}$$

• $$i_{d\_sat}= i_{d\_mtpa}= 0$$

• $$i_{q\_sat}=\text{sat}(i_{q\_mtpa}, i_{\text{max}})$$

If ${\omega }_m> {\omega }_{base}$:

• $i_{d\_fw}= \frac{({\omega }_{e\_base}- {\omega }_e){\lambda }_{pm}}{{\omega }_e{L}_d}$

• $i_{d\_sat}=\text{max}(i_{d\_fw}, -i_{\text{max}})$

• $i_{q\_fw}= \frac{T^{ref}}{\frac{3}{2}\cdot p\cdot {\lambda }_{pm}}$

• $i_{q\_lim}= \sqrt{i_{max}^2- i_{d\_sat}^2}$

• $$i_{q\_sat}=\text{sat}(i_{q\_fw}, i_{\text{q}\_\text{lim}})$$

The saturation function to compute $i_{q\_sat}$ is:

If $i_{q\_fw}< -i_{q\_lim}$,

$i_{q\_sat}= -i_{q\_lim}$

If $i_{q\_fw}>i_{q\_lim}$,

$i_{q\_sat}= i_{q\_lim}$

If $$-i_{q\_lim}\le i_{q\_fw}\ge i_{q\_lim}$$,

$i_{q\_sat}= i_{q\_fw}$

The block outputs these values:

$$I_d^{ref}=i_{d\_sat}$$

$I_q^{ref}=i_{q\_sat}$

where:

• ${\omega }_e$ is the electrical speed corresponding to frequency of stator voltages (radians/ sec).

• ${\omega }_m$ is the rotor mechanical speed (radians/ sec).

• ${\omega }_{base}$ is the mechanical base speed of the motor (radians/ sec).

• ${\omega }_{e\_base}$ is the electrical base speed of the motor (radians/ sec).

• $i_{d\_mtpa}$ is the d-axis phase current corresponding to MTPA (amperes).

• $i_{q\_mtpa}$ is the q-axis phase current corresponding to MTPA (amperes).

• $T^{ref}$ is the reference torque (Nm).

• $$p$$ is the number of motor pole pairs.

• ${\lambda }_{pm}$ is the permanent magnet flux linkage (weber).

• $i_{d\_fw}$ is the d-axis field weakening current (amperes).

• $i_{q\_fw}$ is the q-axis field weakening current (amperes).

• $L_d$ is the d-axis winding inductance (henry).

• $i_{max}$ is the maximum phase current (peak) of the motor (amperes).

• $i_{d\_sat}$ is the d-axis saturation current (amperes).

• $i_{q\_sat}$ is the q-axis saturation current (amperes).

• $$I_d^{ref}$$ is the d-axis current corresponding to the reference torque and reference speed (amperes).

• $I_q^{ref}$ is the q-axis current corresponding to the reference torque and reference speed (amperes).

Interior PMSM

For an interior PMSM, you can achieve maximum torque by computing the d-axis and q-axis reference currents from the torque equation. For the field-weakening operation, the block computes the reference d-axis current using the voltage and current limited maximum torque control (VCLMT) algorithm.

The reference currents for MTPA and field-weakening operations are defined by these equations:

$$i_{m\_ref}= \frac{2\cdot T^{ref}}{3\cdot p\cdot {\lambda }_{pm}}$$

$$i_m=\min (i_{m\_ref}, i_{\text{max}})$$

$i_{d\_mtpa}=\frac{{\lambda }_{pm}}{4\left(L_q-L_d\right)}-\sqrt{\frac{{\lambda }_{pm}^2}{16{\left(L_q-L_d\right)}^2}+\frac{i_m^2}{2}}$

$i_{q\_mtpa}=\sqrt{i_m^2-{\left(i_{d\_mtpa}\right)}^2}$

$$v_{do}=-{\omega }_e{L}_q{i}_q$$

$v_{qo}={\omega }_e\left(L_d{i}_d+ {\lambda }_{pm}\right)$

$$v_{do}^2+v_{qo}^2=v_{\max }^2$$

${\left(L_q{i}_q\right)}^2+{\left(L_d{i}_d+ {\lambda }_{pm}\right)}^2\le \frac{v_{max}^2}{{\omega }_e^2}$

$i_q= \sqrt{i_{max}^2- i_d^2}$

$\left(L_d^2- L_q^2\right)i_d^2+2{\lambda }_{pm}{L}_d{i}_d+ {\lambda }_{pm}^2+ L_q^2{i}_{max}^2-\frac{v_{max}^2}{{\omega }_e^2}=0$

These two equations describe the computation of field-weakening currents for maximum possible torque corresponding to a given speed value:

$i_{d\_fw}= \frac{-{\lambda }_{pm}{L}_d+ \sqrt{{\left({\lambda }_{pm}{L}_d\right)}^2- \left(L_d^2- L_q^2\right)\left({\lambda }_{pm}^2+ L_q^2{i}_{max}^2- \frac{v_{max}^2}{{\omega }_e^2} \right)}}{\left(L_d^2- L_q^2\right)}$

$i_{q\_fw}= \sqrt{i_{max}^2- i_{d\_fw}^2}$

These two equations describe the computation of field-weakening currents for a given speed and torque value.

$9p^2{\left(L_d-L_q\right)}^2{L}_q^2{\omega }_e{}^2{i}_{q\_fw}^4+\left(9p^2{\lambda }_{pm}^2{L}_q^2{\omega }_e{}^2-9p^2{\left(L_d-L_q\right)}^2{v}_{max}^2\right)i_{q\_fw}^2-12T^{ref}p{\lambda }_{pm}{L}_d{L}_q{\omega }_e{}^2{i}_{q\_fw}+4{\left(T^{ref}\right)}^2{L}_d^2{\omega }_e{}^2=0$

$i_{d\_fw}=-\frac{{\lambda }_{pm}}{L_d}+\frac{1}{L_d}\sqrt{\frac{v_{max}^2}{{\omega }_e{}^2}-{\left(L_q{i}_{q\_fw}\right)}^2}$

To reduce the computation time, the block uses approximations to solve the preceding polynomial.

If ${\omega }_m\le {\omega }_{base}$,

$$I_d^{ref}= i_{d\_mtpa}$$

$$I_q^{ref}= i_{q\_mtpa}$$

If ${\omega }_m> {\omega }_{base}$,

$$I_d^{ref}=\max (i_{d\_fw},-i_{\max })$$

$i_{q\_fw}= \sqrt{i_{max}^2- i_{d\_fw}^2}$

If $i_{q\_fw}<i_m$,

$$I_q^{ref}=i_{q\_fw}$$

If $i_{q\_fw}\ge i_m$,

$$I_q^{ref}=i_m$$

For negative reference torque values, the block updates the sign of $$i_m$$ and $I_q^{ref}$ and modifies the equations accordingly.

where:

• $$i_{m\_ref}$$ is the estimated maximum current to produce the reference torque (amperes).

• $$i_m$$ is the saturated value of estimated maximum current (amperes).

• $$i_{d\_\max }$$ is the maximum d-axis phase current (peak) (amperes).

• $$i_{q\_\max }$$ is the maximum q-axis phase current (peak) (amperes).

• $T^{ref}$ is the reference torque (Nm).

• $$I_d^{ref}$$ is the d-axis current component corresponding to the reference torque and reference speed (amperes).

• $I_q^{ref}$ is the q-axis current component corresponding to the reference torque and reference speed (amperes).

• $$p$$ is the number of motor pole pairs.

• ${\lambda }_{pm}$ is the permanent magnet flux linkage (weber).

• $i_{d\_mtpa}$ is the d-axis phase current corresponding to MTPA (amperes).

• $i_{q\_mtpa}$ is the q-axis phase current corresponding to MTPA (amperes).

• $L_d$ is the d-axis winding inductance (henry).

• $L_q$ is the q-axis winding inductance (henry).

• $i_{max}$ is the maximum phase current (peak) of the motor (amperes).

• $v_{max}$ is the maximum fundamental line to neutral voltage (peak) supplied to the motor (volts).

• $v_{do}$ is the d-axis voltage when $i_d$ is zero (volts).

• $v_{qo}$ is the q-axis voltage when $i_q$ is zero (volts).

• ${\omega }_e$ is the electrical speed corresponding to frequency of stator voltages (radians/ sec).

• $i_d$ is the d-axis current (amperes).

• $i_q$ is the q-axis current (amperes).

• $i_{d\_fw}$ is the d-axis field weakening current (amperes).

• $i_{q\_fw}$ is the q-axis field weakening current (amperes).

• ${\omega }_{base}$ is the mechanical base speed of the motor (radians/ sec).

Examples

Field-Weakening Control (with MTPA) of PMSM

Implements the field-oriented control (FOC) technique to control the torque and speed of a three-phase permanent magnet synchronous motor (PMSM). The FOC algorithm requires rotor position feedback, which is obtained by a quadrature encoder sensor. For details about FOC, see Field-Oriented Control (FOC).

• Open Model

Ports

Input

T^ref — Reference torque value

scalar

Reference torque input value for which the block computes the reference current. The unit of this port depends on the Input units parameter value. If you set this parameter to SI Units, the port accepts input signal in Nm.

Data Types: single | double | fixed point

⍵_m — Mechanical speed

scalar

Reference mechanical speed value for which the block computes the reference current. The unit of this port depends on the Input units parameter value. If you set this parameter to SI Units, the port accepts input signal in mechanical radians/sec.

Data Types: single | double | fixed point

V_dc — DC bus voltage

scalar

Variable DC bus voltage. The unit of this port depends on the Input units parameter value. If you set this parameter to SI Units, the port accepts input signal in volts.

Dependencies

To enable this port, set Type of motor to Interior PMSM and Vdc input method to Input port.

Data Types: single | double | fixed point

Output

I_d^ref — Reference d-axis current

scalar

Reference d-axis phase current that can efficiently generate the input torque and speed values.

The unit of this port depends on the block input units defined by the Input units parameter. If you set this parameter to SI Units, the port outputs signal in amperes.

Data Types: single | double | fixed point

I_q^ref — Reference q-axis current

scalar

Reference q-axis phase current that can efficiently generate the input torque and speed values.

The unit of this port depends on the block input units defined by the Input units parameter. If you set this parameter to SI Units, the port outputs signal in amperes.

Data Types: single | double | fixed point

Parameters

Motor Parameters

Type of motor — Type of PMSM

Interior PMSM (default) | Surface PMSM

Type of PMSM based on the location of the permanent magnets.

Number of pole pairs — Number of available pole pairs

4 (default) | scalar

Number of pole pairs available in the motor.

Stator resistance per phase (Ohm) — Resistance of stator phase winding (ohms)

0.36 (default) | scalar

Resistance of the stator phase winding (in ohms).

Dependencies

To enable this parameter, set Type of motor to Interior PMSM.

Stator d-axis inductance (H) — d-axis stator winding inductance

0.2e-3 (default) | scalar

Stator winding inductance (in henry) along the d-axis of the rotating dq reference frame.

Stator q-axis inductance (H) — q-axis stator winding inductance

0.4e-3 (default) | scalar

Stator winding inductance (in henry) along the q-axis of the rotating dq reference frame.

Dependencies

To enable this parameter, set Type of motor to Interior PMSM.

Permanent magnet flux linkage (Wb) — Magnetic flux linkage of permanent magnets

6.4e-3 (default) | scalar

Magnetic flux linkage (in weber) between the stator windings and permanent magnets on the rotor (weber).

Max current (A) — Maximum phase current limit for motor (amperes)

7.1 (default) | scalar

Maximum phase current limit for the motor (in amperes).

Vdc input method — Input method for Vdc

Specify via dialog (default) | Input port

Use one of these methods to specify the Vdc input.

• Specify via dialog — Specify the fixed reference voltage Vdc using the parameter DC voltage (V).

• Input port — Specify the variable reference voltage Vdc using the input port Vdc.

  Note

  If you select the Input port input method for Vdc, ensure that you configure the Base voltage (V) parameter in the Input Units tab.

Dependencies

To enable this parameter, set Type of motor to Interior PMSM.

DC voltage (V) — DC bus voltage (volts)

24 (default) | scalar

DC bus voltage (in volts).

Dependencies

To enable this parameter, set Type of motor to Interior PMSM and Vdc input method to Specify via dialog.

Input Units

Input signal units — Unit of block input values

Per-Unit (PU) (default) | SI Units

Unit of the block input values.

Base speed (rpm) — Base speed of motor (rpm)

4107 (default) | scalar

Speed of the motor (in rpm) at the rated voltage and rated current outside the field-weakening region.

Dependencies

To enable this parameter, set Input signal units to Per-Unit (PU).

Base current (A) — Base current for per-unit conversion (amperes)

19.3 (default) | scalar

Current (in amperes) corresponding to 1 per-unit. We recommend that you use the maximum current detected by an analog to digital converter (ADC) as the base current.

Dependencies

To enable this parameter, set Input signal units to Per-Unit (PU).

Base torque (Nm) — Base torque for per-unit conversion (Nm)

0.74112 (default) | scalar

Torque (in Nm) corresponding to 1 per-unit. See Per-Unit System page for more details.

You cannot specify this value. The block computes this value internally using other parameters.

Dependencies

To display this parameter, set Input signal units to Per-Unit (PU).

Base voltage (V) — Base voltage for per-unit conversion (volts)

24/sqrt(3) (default) | scalar

Voltage (in volts) corresponding to 1 per-unit. See Per-Unit System page for more details.

You cannot specify this value. The block computes this value internally using other parameters.

Dependencies

To display this parameter, set Input signal units to Per-Unit (PU).

Implementation

Prioritise execution time — Scale down computations to reduce algorithm execution time

on (default) | off

If you select this parameter, the block follows computational approximations to prioritize and reduce the algorithm execution time. Clear this field to run the block algorithms iteratively without using approximations.

References

[1] B. Bose, Modern Power Electronics and AC Drives. Prentice Hall, 2001. ISBN-0-13-016743-6.

[2] Morimoto, Shigeo, Masayuka Sanada, and Yoji Takeda. "Wide-speed operation of interior permanent magnet synchronous motors with high-performance current regulator." IEEE Transactions on Industry Applications, Vol. 30, Issue 4, July/August 1994, pp. 920-926.

[3] Li, Muyang. "Flux-Weakening Control for Permanent-Magnet Synchronous Motors Based on Z-Source Inverters." Master's Thesis, Marquette University, e-Publications@Marquette, Fall 2014.

[4] Briz, Fernando, Michael W. Degner, and Robert D. Lorenz. "Analysis and design of current regulators using complex vectors." IEEE Transactions on Industry Applications, Vol. 36, Issue 3, May/June 2000, pp. 817-825.

[5] Lorenz, Robert D., Thomas Lipo, and Donald W. Novotny. "Motion control with induction motors." Proceedings of the IEEE, Vol. 82, Issue 8, August 1994, pp. 1215-1240.

[6] Briz, Fernando, et al. "Current and flux regulation in field-weakening operation [of induction motors]." IEEE Transactions on Industry Applications, Vol. 37, Issue 1, Jan/Feb 2001, pp. 42-50.

[7] TI Application Note, "Sensorless-FOC With Flux-Weakening and MTPA for IPMSM Motor Drives."

[8] Haque, M. E., L. Zhong, and M. F. Rahman. "Improved trajectory control for an interior permanent magnet synchronous motor drive with extended operating limit." Journal of Electrical & Electronics Engineering. Vol. 22, Number 1, 2003, p. 49.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.

Version History

Introduced in R2020a