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 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 of the rotor and outputs the reference d and q axis phase 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.
$v_{q}$ is the q-axis voltage.
$i_{d}$ is the d-axis current.
$i_{q}$ is the q-axis current.
$R_{s}$ is the stator phase winding resistance.
$λ_{p m}$ is the permanent magnet flux linkage.
$λ_{d}$ is the d-axis flux linkage.
$λ_{q}$ is the q-axis flux linkage.
$ω_{e}$ is the rotor electrical speed.
$ω_{m}$ is the rotor mechanical speed.
$L_{d}$ is the d-axis winding inductance.
$L_{q}$ is the q-axis winding inductance.
$T_{e}$ is the electromechanical torque produced by PMSM.
$T_{L}$ is the load torque.
$p$ is the number of motor pole pairs.
$J$ is the inertia coefficient.
$B$ is the friction coefficient.

Base Speed

These equations describe the computation of the motor base speed:

If $ω_{m} \leq ω_{b a s e}$ , $i_{d}$ is zero. Therefore, we compute the d-axis and q-axis voltages as:

$v_{d o} = - ω_{e} L_{q} i_{q}$

$v_{q o} = i_{q} R_{s} + ω_{e} λ_{p m}$

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

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

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

where:

$ω_{e}$ is the rotor electrical speed.
$ω_{b a s e}$ is the maximum motor speed outside field weakening region.
$i_{d}$ is the d-axis current.
$i_{q}$ is the q-axis current.
$v_{d o}$ is the d-axis voltage when $i_{d}$ is zero.
$v_{q o}$ is the q-axis voltage when $i_{d}$ is zero.
$L_{d}$ is the d-axis winding inductance.
$L_{q}$ is the q-axis winding inductance.
$R_{s}$ is the stator phase winding resistance.
$λ_{p m}$ is the permanent magnet flux linkage.
$v_{d}$ is the d-axis voltage.
$v_{q}$ is the q-axis voltage.
$v_{m a x}$ is the maximum voltage that the inverter can provide.
$v_{d c}$ is the DC bus voltage.
$i_{m a x}$ is the maximum current that the inverter can provide.

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 rotor electrical speed.
$ω_{m}$ is the rotor mechanical speed.
$ω_{b a s e}$ is the maximum motor speed outside field weakening region.
$ω_{e_b a s e}$ is the maximum electrical speed of motor outside the field weakening region.
$i_{d_m t p a}$ is the d-axis phase current MTPA.
$i_{q_m t p a}$ is the q-axis phase current MTPA.
$T^{r e f}$ is the reference torque (block input).
$p$ is the number of motor pole pairs.
$λ_{p m}$ is the permanent magnet flux linkage.
$i_{d_f w}$ is the d-axis field weakening current.
$i_{q_f w}$ is the q-axis field weakening current.
$L_{d}$ is the d-axis winding inductance.
$i_{m a x}$ is the maximum current that the inverter can provide.
$i_{d_s a t}$ is the d-axis saturation current.
$i_{q_s a t}$ is the q-axis saturation current.
$I_{d}^{r e f}$ is the d-axis current component corresponding to the reference torque and reference speed (block inputs).
$I_{q}^{r e f}$ is the q-axis current component corresponding to the reference torque and reference speed (block inputs).

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.

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

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

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

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

$v_{d o} = - ω_{e} L_{q} i_{q}$

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

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

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

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

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

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

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

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.
$i_{m}$ is the saturated value of estimated maximum current.
$i_{d_\max}$ is the maximum permissible d-axis current.
$i_{q_\max}$ is the maximum permissible q-axis current.
$T^{r e f}$ is the reference torque (block input).
$I_{d}^{r e f}$ is the d-axis current component corresponding to the reference torque and reference speed (block inputs).
$I_{q}^{r e f}$ is the q-axis current component corresponding to the reference torque and reference speed (block inputs).
$p$ is the number of motor pole pairs.
$λ_{p m}$ is the permanent magnet flux linkage.
$i_{d_m t p a}$ is the d-axis phase current MTPA.
$i_{q_m t p a}$ is the q-axis phase current MTPA.
$L_{d}$ is the d-axis winding inductance.
$L_{q}$ is the q-axis winding inductance.
$i_{m a x}$ is the maximum current that the inverter can provide.
$v_{m a x}$ is the maximum voltage that the inverter can provide.
$v_{d o}$ is the d-axis voltage when $i_{d}$ is zero.
$v_{q o}$ is the q-axis voltage when $i_{d}$ is zero.
$ω_{e}$ is the rotor electrical speed.
$i_{d}$ is the d-axis current.
$i_{q}$ is the q-axis current.
$i_{d_f w}$ is the d-axis field weakening current.
$i_{q_f w}$ is the q-axis field weakening current.
$ω_{b a s e}$ is the maximum motor speed outside the field weakening region.

Ports

Input

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`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` — Reference feedback speed
scalar

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

Data Types: single | double | fixed point

Output

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`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

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`Type of motor` — Type of PMSM based on location of permanent magnets
`Interior PMSM` (default) | `Surface PMSM`

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

`Number of pole pairs` — Number of pole pairs available in motor
`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 rotor's permanent magnets
`6.4e-3` (default) | scalar

Magnetic flux linkage between the stator windings and the rotor's permanent magnets (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 you to use the maximum current detected by Analog to Digital Converter (ADC) as the base current.

Dependencies

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

Documentation

MTPA Control Reference

Description

Mathematical Model of PMSM

Base Speed

Surface PMSM

Interior PMSM

Ports

Input

`T^ref` — Reference torque value
scalar

`⍵_m` — Reference feedback 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 based on location of permanent magnets
`Interior PMSM` (default) | `Surface PMSM`

`Number of pole pairs` — Number of pole pairs available in motor
`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 rotor's 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

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 — Reference feedback speed scalar

Output

Idref — Reference d-axis current scalar

Iqref — Reference q-axis current scalar

Parameters

Type of motor — Type of PMSM based on location of permanent magnets Interior PMSM (default) | Surface PMSM

Number of pole pairs — Number of pole pairs available in motor 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 rotor's 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

Motor Control Blockset Documentation

Support

`T^ref` — Reference torque value
scalar

`⍵_m` — Reference feedback speed
scalar

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

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

`Type of motor` — Type of PMSM based on location of permanent magnets
`Interior PMSM` (default) | `Surface PMSM`

`Number of pole pairs` — Number of pole pairs available in motor
`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 rotor's 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