MTPA Control Reference

Compute reference currents for Maximum Torque Per Ampere (MTPA) and field-weakening operation

Library:
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 results in efficient output for the permanent magnet synchronous motor (PMSM).

The block accepts the reference torque and feedback mechanical speed and outputs the corresponding d- and q-axes reference current values for MTPA and field-weakening operations.

The block computes the reference current values by solving mathematical relationships. The calculations use SI unit system. When working with the Per-Unit (PU) system, the block converts PU input signals to SI units to perform computations, and converts them back to PU values at the output.

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 dynamics of PMSM in the rotor flux reference frame:

$v_{d} = i_{d} R_{s} + \frac{d λ_{d}}{d t} - ω_{e} L_{q} i_{q}$

$v_{q} = i_{q} R_{s} + \frac{d λ_{q}}{d t} + ω_{e} L_{d} i_{d} + ω_{e} λ_{p m}$

$λ_{d} = L_{d} i_{d} + λ_{p m}$

$λ_{q} = L_{q} i_{q}$

$T_{e} = \frac{3}{2} p (λ_{p m} i_{q} + (L_{d} - L_{q}) i_{d} i_{q})$

$T_{e} - T_{L} = J \frac{d ω_{m}}{d t} + B ω_{m}$

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).
$λ_{p m}$ is the permanent magnet flux linkage (Weber).
$λ_{d}$ is the d-axis flux linkage (Weber).
$λ_{q}$ is the q-axis flux linkage (Weber).
$ω_{e}$ is the electrical speed corresponding to frequency of stator voltages (Radians/ sec).
$ω_{m}$ 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:

$v_{d o} = - ω_{e} L_{q} i_{q}$

$v_{q o} = ω_{e} (L_{d} i_{d} + λ_{p m})$

$v_{m a x} = \frac{v_{d c}}{\sqrt{3}} - R_{s} i_{m a x} \geq \sqrt{v_{d o}^{2} + v_{q o}^{2}}$

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

$i_{m a x}^{2} = i_{d}^{2} + i_{q}^{2}$

In the preceding equation, $i_{d}$ is zero for surface PMSMs. For interior PMSMs, values of $i_{d}$ and $i_{q}$ corresponding to MTPA are considered.

Using the preceding relationships, we can compute the base speed as:

$ω_{b a s e} = \frac{1}{p} \cdot \frac{v_{m a x}}{\sqrt{{(L_{q} i_{q})}^{2} + {(L_{d} i_{d} + λ_{p m})}^{2}}}$

where:

$ω_{e}$ is the electrical speed corresponding to frequency of stator voltages (Radians/ sec).
$ω_{b a s e}$ 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).
$v_{d o}$ is the d-axis voltage when $i_{d}$ is zero (Volts).
$v_{q o}$ 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).
$λ_{p m}$ is the permanent magnet flux linkage (Weber).
$v_{d}$ is the d-axis voltage (Volts).
$v_{q}$ is the q-axis voltage (Volts).
$v_{m a x}$ is the maximum fundamental line to neutral voltage (peak) supplied to the motor (Volts).
$v_{d c}$ is the dc voltage supplied to the inverter (Volts).
$i_{m a x}$ 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 field-weakening operation, the reference d-axis current is computed by constant-voltage-constant-power control (CVCP) algorithm defined by these equations:

If $ω_{m} \leq ω_{b a s e}$ :

$i_{d_m t p a} = 0$
$i_{q_m t p a} = \frac{T^{r e f}}{\frac{3}{2} \cdot p \cdot λ_{p m}}$
$i_{d_s a t} = i_{d_m t p a} = 0$
$i_{q_s a t} = sat (i_{q_m t p a}, i_{max})$

If $ω_{m} > ω_{b a s e}$ :

$i_{d_f w} = \frac{(ω_{e_b a s e} - ω_{e}) λ_{p m}}{ω_{e} L_{d}}$
$i_{d_s a t} = max (i_{d_f w}, - i_{max})$
$i_{q_f w} = \frac{T^{r e f}}{\frac{3}{2} \cdot p \cdot λ_{p m}}$
$i_{q_l i m} = \sqrt{i_{m a x}^{2} - i_{d_s a t}^{2}}$
$i_{q_s a t} = sat (i_{q_f w}, i_{q_lim})$

The saturation function used to compute $i_{q_s a t}$ is described below:

If $i_{q_f w} < - i_{q_l i m}$ ,

$i_{q_s a t} = - i_{q_l i m}$

If $i_{q_f w} > i_{q_l i m}$ ,

$i_{q_s a t} = i_{q_l i m}$

If $- i_{q_l i m} \leq i_{q_f w} \geq i_{q_l i m}$ ,

$i_{q_s a t} = i_{q_f w}$

The block outputs the following values:

$I_{d}^{r e f} = i_{d_s a t}$

$I_{q}^{r e f} = i_{q_s a t}$

where:

$ω_{e}$ is the electrical speed corresponding to frequency of stator voltages (Radians/ sec).
$ω_{m}$ is the rotor mechanical speed (Radians/ sec).
$ω_{b a s e}$ is the mechanical base speed of the motor (Radians/ sec).
$ω_{e_b a s e}$ is the electrical base speed of the motor (Radians/ sec).
$i_{d_m t p a}$ is the d-axis phase current corresponding to MTPA (Amperes).
$i_{q_m t p a}$ is the q-axis phase current corresponding to MTPA (Amperes).
$T^{r e f}$ is the reference torque (Nm).
$p$ is the number of motor pole pairs.
$λ_{p m}$ is the permanent magnet flux linkage (Weber).
$i_{d_f w}$ is the d-axis field weakening current (Amperes).
$i_{q_f w}$ is the q-axis field weakening current (Amperes).
$L_{d}$ is the d-axis winding inductance (Henry).
$i_{m a x}$ is the maximum phase current (peak) of the motor (Amperes).
$i_{d_s a t}$ is the d-axis saturation current (Amperes).
$i_{q_s a t}$ is the q-axis saturation current (Amperes).
$I_{d}^{r e f}$ is the d-axis current corresponding to the reference torque and reference speed (Amperes).
$I_{q}^{r e f}$ 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 field-weakening operation, the reference d-axis current is computed by voltage and current limited maximum torque control (VCLMT) algorithm.

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

$i_{m_r e f} = \frac{2 \cdot T^{r e f}}{3 \cdot p \cdot λ_{p m}}$

$i_{m} = \max (i_{m_r e f}, i_{max})$

$i_{d_m t p a} = \frac{λ_{p m}}{4 (L_{q} - L_{d})} - \sqrt{\frac{λ_{p m}^{2}}{16 {(L_{q} - L_{d})}^{2}} + \frac{i_{m}^{2}}{2}}$

$i_{q_m t p a} = \sqrt{i_{m}^{2} - {(i_{d_m t p a})}^{2}}$

$v_{d o} = - ω_{e} L_{q} i_{q}$

$v_{q o} = ω_{e} (L_{d} i_{d} + λ_{p m})$

$v_{d o}^{2} + v_{q o}^{2} = v_{\max}^{2}$

${(L_{q} i_{q})}^{2} + {(L_{d} i_{d} + λ_{p m})}^{2} \leq \frac{v_{m a x}^{2}}{ω_{e}^{2}}$

$i_{q} = \sqrt{i_{m a x}^{2} - i_{d}^{2}}$

$(L_{d}^{2} - L_{q}^{2}) i_{d}^{2} + 2 λ_{p m} L_{d} i_{d} + λ_{p m}^{2} + L_{q}^{2} i_{m a x}^{2} - \frac{v_{m a x}^{2}}{ω_{e}^{2}} = 0$

$i_{d_f w} = \frac{- λ_{p m} L_{d} + \sqrt{{(λ_{p m} L_{d})}^{2} - (L_{d}^{2} - L_{q}^{2}) (λ_{p m}^{2} + L_{q}^{2} i_{m a x}^{2} - \frac{v_{m a x}^{2}}{ω_{e}^{2}})}}{(L_{d}^{2} - L_{q}^{2})}$

$i_{q_f w} = \sqrt{i_{m a x}^{2} - i_{d_f w}^{2}}$

If $ω_{m} \leq ω_{b a s e}$ ,

$I_{d}^{r e f} = i_{d_m t p a}$

$I_{q}^{r e f} = i_{q_m t p a}$

If $ω_{m} > ω_{b a s e}$ ,

$I_{d}^{r e f} = \max (i_{d_f w}, - i_{\max})$

$i_{q_f w} = \sqrt{i_{m a x}^{2} - i_{d_f w}^{2}}$

If $i_{q_f w} < i_{m}$ ,

$I_{q}^{r e f} = i_{q_f w}$

If $i_{q_f w} \geq i_{m}$ ,

$I_{q}^{r e f} = i_{m}$

For negative reference torque values, the sign of $i_{m}$ and $I_{q}^{r e f}$ are updated and equations are modified accordingly.

where:

$i_{m_r e f}$ 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).
$T^{r e f}$ is the reference torque (Nm).
$I_{d}^{r e f}$ is the d-axis current component corresponding to the reference torque and reference speed (Amperes).
$I_{q}^{r e f}$ is the q-axis current component corresponding to the reference torque and reference speed (Amperes).
$p$ is the number of motor pole pairs.
$λ_{p m}$ is the permanent magnet flux linkage (Weber).
$i_{d_m t p a}$ is the d-axis phase current corresponding to MTPA (Amperes).
$i_{q_m t p a}$ 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).
$i_{m a x}$ is the maximum phase current (peak) of the motor (Amperes).
$v_{m a x}$ is the maximum fundamental line to neutral voltage (peak) supplied to the motor (Volts).
$v_{d o}$ is the d-axis voltage when $i_{d}$ is zero (Volts).
$v_{q o}$ is the q-axis voltage when $i_{q}$ is zero (Volts).
$ω_{e}$ 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).
$i_{d_f w}$ is the d-axis field weakening current (Amperes).
$i_{q_f w}$ is the q-axis field weakening current (Amperes).
$ω_{b a s e}$ is the mechanical base speed of the motor (Radians/ sec).

Ports

Input

expand all

`T^ref` — Reference torque value
scalar

Reference torque input value for which the block computes the reference current.

Data Types: single | double | fixed point

`⍵_m` — Mechanical speed
scalar

Reference mechanical speed value for which the block computes the reference current.

Data Types: single | double | fixed point

Output

expand all

`I_d^ref` — Reference d-axis current
scalar

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

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.

Data Types: single | double | fixed point

Parameters

expand all

`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 (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 (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 (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 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 (amperes).

`DC voltage (V)` — DC bus voltage (volts)
`24` (default) | scalar

DC bus voltage (volts)

Dependencies

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

`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)
`3182` (default) | scalar

Speed of the motor at the rated voltage and rated current outside the field weakening region.

`Base current (A)` — Base current for per-unit conversion (amperes)
`19.3` (default) | scalar

Current 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).

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."

Documentation

MTPA Control Reference

Description

Mathematical Model of PMSM

Base Speed

Surface PMSM

Interior PMSM

Ports

Input

`T^ref` — Reference torque value
scalar

`⍵_m` — Mechanical speed
scalar

Output

`I_d^ref` — Reference d-axis current
scalar

`I_q^ref` — Reference q-axis current
scalar

Parameters

`Type of motor` — Type of PMSM
`Interior PMSM` (default) | `Surface PMSM`

`Number of pole pairs` — Number of available pole pairs
`4` (default) | scalar

`Stator resistance per phase (Ohm)` — Resistance of stator phase winding (ohms)
`0.36` (default) | scalar

Dependencies

`Stator d-axis inductance (H)` — d-axis stator winding inductance
`0.2e-3` (default) | scalar

`Stator q-axis inductance (H)` — q-axis stator winding inductance
`0.4e-3` (default) | scalar

Dependencies

`Permanent magnet flux linkage (Wb)` — Magnetic flux linkage of permanent magnets
`6.4e-3` (default) | scalar

`Max current (A)` — Maximum phase current limit for motor (amperes)
`7.1` (default) | scalar

`DC voltage (V)` — DC bus voltage (volts)
`24` (default) | scalar

Dependencies

`Input signal units` — Unit of block input values
`Per-Unit (PU)` (default) | `SI Units`

`Base speed (rpm)` — Base speed of motor (rpm)
`3182` (default) | scalar

`Base current (A)` — Base current for per-unit conversion (amperes)
`19.3` (default) | scalar

Dependencies

References

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™.

See Also

Topics

Motor Control Blockset Documentation

Support

Documentation

MTPA Control Reference

Description

Mathematical Model of PMSM

Base Speed

Surface PMSM

Interior PMSM

Ports

Input

Tref — Reference torque value scalar

⍵m — Mechanical speed scalar

Output

Idref — Reference d-axis current scalar

Iqref — Reference q-axis current scalar

Parameters

Type of motor — Type of PMSM Interior PMSM (default) | Surface PMSM

Number of pole pairs — Number of available pole pairs 4 (default) | scalar

Stator resistance per phase (Ohm) — Resistance of stator phase winding (ohms) 0.36 (default) | scalar

Dependencies

Stator d-axis inductance (H) — d-axis stator winding inductance 0.2e-3 (default) | scalar

Stator q-axis inductance (H) — q-axis stator winding inductance 0.4e-3 (default) | scalar

Dependencies

Permanent magnet flux linkage (Wb) — Magnetic flux linkage of permanent magnets 6.4e-3 (default) | scalar

Max current (A) — Maximum phase current limit for motor (amperes) 7.1 (default) | scalar

DC voltage (V) — DC bus voltage (volts) 24 (default) | scalar

Dependencies

Input signal units — Unit of block input values Per-Unit (PU) (default) | SI Units

Base speed (rpm) — Base speed of motor (rpm) 3182 (default) | scalar

Base current (A) — Base current for per-unit conversion (amperes) 19.3 (default) | scalar

Dependencies

References

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™.

See Also

Topics

Motor Control Blockset Documentation

Support

`T^ref` — Reference torque value
scalar

`⍵_m` — Mechanical speed
scalar

`I_d^ref` — Reference d-axis current
scalar

`I_q^ref` — Reference q-axis current
scalar

`Type of motor` — Type of PMSM
`Interior PMSM` (default) | `Surface PMSM`

`Number of pole pairs` — Number of available pole pairs
`4` (default) | scalar

`Stator resistance per phase (Ohm)` — Resistance of stator phase winding (ohms)
`0.36` (default) | scalar

`Stator d-axis inductance (H)` — d-axis stator winding inductance
`0.2e-3` (default) | scalar

`Stator q-axis inductance (H)` — q-axis stator winding inductance
`0.4e-3` (default) | scalar

`Permanent magnet flux linkage (Wb)` — Magnetic flux linkage of permanent magnets
`6.4e-3` (default) | scalar

`Max current (A)` — Maximum phase current limit for motor (amperes)
`7.1` (default) | scalar

`DC voltage (V)` — DC bus voltage (volts)
`24` (default) | scalar

`Input signal units` — Unit of block input values
`Per-Unit (PU)` (default) | `SI Units`

`Base speed (rpm)` — Base speed of motor (rpm)
`3182` (default) | scalar

`Base current (A)` — Base current for per-unit conversion (amperes)
`19.3` (default) | scalar

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™.