Surface Mount PMSM

Three-phase exterior permanent magnet synchronous motor with sinusoidal back electromotive force

Libraries:
Powertrain Blockset / Propulsion / Electric Motors and Inverters
Motor Control Blockset / Electrical Systems / Motors

Description

The Surface Mount PMSM block implements a three-phase exterior permanent magnet synchronous motor (PMSM) with sinusoidal back electromotive force. The block uses the three-phase input voltages to regulate the individual phase currents, allowing control of the motor torque or speed.

By default, the block sets the Simulation type parameter to Continuous to use a continuous sample time during simulation. If you want to generate code for fixed-step double- and single-precision targets, considering setting the parameter to Discrete. Then specify a Sample Time, Ts parameter.

On the Parameters tab, if you select Back-emf constant (Ke) or Torque constant (Kt), the block implements one of these equations to calculate the permanent flux linkage constant.

Setting	Equation
`Back-emf constant (Ke)`	$λ_{p m} = \frac{1}{\sqrt{3}} \cdot \frac{K_{e}}{1000 P} \cdot \frac{60}{2 π}$
`Torque constant (Kt)`	$λ_{p m} = \frac{2}{3} \cdot \frac{K_{t}}{P}$

Motor Construction

This figure shows the motor construction with a single pole pair on the rotor.

PMSM surface mount motor with a two-pole rotor

The motor magnetic field due to the permanent magnets creates a sinusoidal rate of change of flux with motor angle.

For the axes convention, the a-phase and permanent magnet fluxes are aligned when motor angle θ_r is zero.

Three-Phase Sinusoidal Model Electrical System

The block implements these equations, expressed in the motor flux reference frame (dq frame). All quantities in the motor reference frame are referred to the stator.

$\begin{array}{l} ω_{e} = P ω_{m} \\ \frac{d}{d t} i_{d} = \frac{1}{L_{d}} v_{d} - \frac{R}{L_{d}} i_{d} + \frac{L_{q}}{L_{d}} P ω_{m} i_{q} \end{array}$

$\frac{d}{d t} i_{q} = \frac{1}{L_{q}} v_{q} - \frac{R}{L_{q}} i_{q} - \frac{L_{d}}{L_{q}} P ω_{m} i_{d} - \frac{λ_{p m} P ω_{m}}{L_{q}}$

$T_{e} = 1.5 P [λ_{p m} i_{q} + (L_{d} - L_{q}) i_{d} i_{q}]$

The L_q and L_d inductances represent the relation between the phase inductance and the motor position due to the saliency of the motor magnets. For the surface mount PMSM, $L_{d} = L_{q}$ .

The equations use these variables.

L_q, L_d	q- and d-axis inductances (H)
R	Resistance of the stator windings (ohm)
i_q, i_d	q- and d-axis currents (A)
v_q, v_d	q- and d-axis voltages (V)
ω_m	Angular mechanical velocity of the motor (rad/s)
ω_e	Angular electrical velocity of the motor (rad/s)
λ_pm	Permanent magnet flux linkage (Wb)
K_e	Back electromotive force (EMF) (Vpk_LL/krpm, where Vpk_LL is the peak voltage line-to-line measurement)
K_t	Torque constant (N·m/A) K_t = Rated torque of motor / Ia_peak_rated, where Ia_peak_rated is the peak amplitude of phase-a at rated current operation.
P	Number of pole pairs
T_e	Electromagnetic torque (Nm)
Θ_e	Electrical angle (rad)

Mechanical System

The motor angular velocity is given by:

$\begin{matrix} \frac{d}{d t} ω_{m} = \frac{1}{J} (T_{e} - T_{f} - F ω_{m} - T_{m}) \\ \frac{d θ_{m}}{d t} = ω_{m} \end{matrix}$

The equations use these variables.

J	Combined inertia of motor and load (kgm^2)
F	Combined viscous friction of motor and load (N·m/(rad/s))
θ_m	Motor mechanical angular position (rad)
T_m	Motor shaft torque (Nm)
T_e	Electromagnetic torque (Nm)
T_f	Motor shaft static friction torque (Nm)
ω_m	Angular mechanical velocity of the motor (rad/s)

Power Accounting

For the power accounting, the block implements these equations.

Bus Signal			Description	Variable	Equations
`PwrInfo`	`PwrTrnsfrd` — Power transferred between blocks Positive signals indicate flow into block Negative signals indicate flow out of block	`PwrMtr`	Mechanical power	P_mot	$P_{m o t} = - ω_{m} T_{e}$
		`PwrBus`	Electrical power	P_bus	$P_{b u s} = v_{a n} i_{a} + v_{b n} i_{b} + v_{c n} i_{c}$
	`PwrNotTrnsfrd` — Power crossing the block boundary, but not transferred Positive signals indicate an input Negative signals indicate a loss	`PwrElecLoss`	Resistive power loss	P_elec	$P_{e l e c} = - \frac{3}{2} (R_{s} i_{s d}^{2} + R_{s} i_{s q}^{2})$
		`PwrMechLoss`	Mechanical power loss	P_mech	When Mechanical input configuration is set to `Torque`: $P_{m e c h} = - (ω_{m}^{2} F + \| ω_{m} \| T_{f})$ When Mechanical input configuration is set to `Speed`: $P_{m e c h} = 0$
	`PwrStored` — Stored energy rate of change Positive signals indicate an increase Negative signals indicate a decrease	`PwrMtrStored`	Stored motor power	P_str	$P_{s t r} = P_{b u s} + P_{m o t} + P_{e l e c} + P_{m e c h}$

The equations use these variables.

R_s	Stator resistance (ohm)
i_a, i_b, i_c	Stator phase a, b, and c current (A)
i_sq, i_sd	Stator q- and d-axis currents (A)
v_an, v_bn, v_cn	Stator phase a, b, and c voltage (V)
ω_m	Angular mechanical velocity of the motor (rad/s)
F	Combined motor and load viscous damping N·m/(rad/s)
T_e	Electromagnetic torque (Nm)
T_f	Combined motor and load friction torque (Nm)

Amplitude Invariant dq Transformation

The block uses these equations to implement amplitude invariant dq transformation to ensure that the dq and three phase amplitudes are equal.

$[\begin{matrix} v_{s d} \\ v_{s q} \end{matrix}] = \frac{2}{3} [\begin{matrix} cos (Θ_{d a}) & cos (Θ_{d a} - \frac{2 π}{3}) & cos (Θ_{d a} + \frac{2 π}{3}) \\ - sin (Θ_{d a}) & - sin (Θ_{d a} - \frac{2 π}{3}) & - sin (Θ_{d a} + \frac{2 π}{3}) \end{matrix}] [\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}]$

$[\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] = [\begin{matrix} cos (Θ_{d a}) & - sin (Θ_{d a}) \\ \begin{matrix} cos (Θ_{d a} - \frac{2 π}{3}) \\ cos (Θ_{d a} + \frac{2 π}{3}) \end{matrix} & \begin{matrix} - sin (Θ_{d a} - \frac{2 π}{3}) \\ - sin (Θ_{d a} + \frac{2 π}{3}) \end{matrix} \end{matrix}] [\begin{matrix} i_{s d} \\ i_{s q} \end{matrix}]$

The equations use these variables.

Θ_da	dq stator electrical angle with respect to the rotor a-axis (rad)
v_sq, v_sd	Stator q- and d-axis voltages (V)
i_sq, i_sd	Stator q- and d-axis currents (A)
v_a, v_b, v_c	Stator voltage phases a, b, c (V)
i_a, i_b, i_c	Stator currents phases a, b, c (A)

Ports

Input

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LdTrq — Load torque on motor
`scalar`

Load torque on the motor shaft, T_m, in N·m.

Dependencies

To create this port, select Torque for the Mechanical input configuration parameter.

Spd — Motor shaft speed
`scalar`

Angular velocity of the motor, ω_m, in rad/s.

Dependencies

To create this port, select Speed for the Mechanical input configuration parameter.

PhaseVolt — Stator terminal voltages
`1`-by-`3` array

Stator terminal voltages, V_a, V_b, and V_c, in V.

Output

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Info — Bus signal
bus

The bus signal contains these block calculations.

Signal			Description	Variable	Units
`IaStator`			Stator phase current A	i_a	A
`IbStator`			Stator phase current B	i_b	A
`IcStator`			Stator phase current C	i_c	A
`IdSync`			Direct axis current	i_d	A
`IqSync`			Quadrature axis current	i_q	A
`VdSync`			Direct axis voltage	v_d	V
`VqSync`			Quadrature axis voltage	v_q	V
`MtrSpd`			Angular mechanical velocity of the motor	ω_m	rad/s
`MtrPos`			Motor mechanical angular position	θ_m	rad
`MtrTrq`			Electromagnetic torque	T_e	N·m
`PwrInfo`	`PwrTrnsfrd`	`PwrMtr`	Mechanical power	P_mot	W
	`PwrTrnsfrd`	`PwrBus`	Electrical power	P_bus	W
	`PwrNotTrnsfrd`	`PwrElecLoss`	Resistive power loss	P_elec	W
	`PwrNotTrnsfrd`	`PwrMechLoss`	Mechanical power loss	P_mech	W
	`PwrStored`	`PwrMtrStored`	Stored motor power	P_str	W

PhaseCurr — Phase a, b, c current
`1`-by-`3` array

Phase a, b, c current, i_a, i_b, and i_c, in A.

MtrTrq — Motor torque
`scalar`

Motor torque, T_mtr, in N·m.

Dependencies

To create this port, select Speed for the Mechanical input configuration parameter.

MtrSpd — Motor speed
`scalar`

Angular speed of the motor, ω_mtr, in rad/s.

Dependencies

To create this port, select Torque for the Mechanical input configuration parameter.

Parameters

expand all

Block Options

Mechanical input configuration — Select port configuration
`Torque` (default) | `Speed`

This table summarizes the port configurations.

Port Configuration	Creates Input Port	Creates Output Port
`Torque`	`LdTrq`	`MtrSpd`
`Speed`	`Spd`	`MtrTrq`

Simulation type — Select simulation type
`Continuous` (default) | `Discrete`

By default, the block uses a continuous sample time during simulation. If you want to generate code for single-precision targets, considering setting the parameter to Discrete.

Dependencies

Setting Simulation type to Discrete creates the Sample Time, Ts parameter.

Sample Time (Ts) — Sample time for discrete integration
`0.001` (default) | `scalar`

Integration sample time for discrete simulation, in s.

Dependencies

Setting Simulation type to Discrete creates the Sample Time, Ts parameter.

Parameters

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

Motor pole pairs, P.

Stator resistance per phase (Rs) — Resistance
`.2` (default) | `scalar`

Stator phase resistance per phase, R_s, in ohm.

Stator d-axis inductance (Ldq_) — Inductance
`3.752e-4` (default) | `scalar`

Stator inductance, L_dq, in H.

Permanent flux linkage constant (lambda_pm) — Flux
`0.1194` (default) | `scalar`

Permanent flux linkage constant, λ_pm, in Wb.

Back-emf constant (Ke) — Back electromotive force
`scalar`

Back electromotive force, EMF, K_e, in peak Vpk_LL/krpm. Vpk_LL is the peak voltage line-to-line measurement.

To calculate the permanent flux linkage constant, the block implements this equation.

$λ_{p m} = \frac{1}{\sqrt{3}} \cdot \frac{K_{e}}{1000 P} \cdot \frac{60}{2 π}$

Torque constant (Kt) — Torque constant
`scalar`

Torque constant, K_t, in N·m/A.

K_t = Rated torque of motor / Ia_peak_rated, where Ia_peak_rated is the peak amplitude of phase-a at rated current operation.

To calculate the permanent flux linkage constant, the block implements this equation.

$λ_{p m} = \frac{2}{3} \cdot \frac{K_{t}}{P}$

Physical inertia, viscous damping, static friction (mechanical) — Inertia, damping, friction
`[0.002700,4.924e-4,0]` (default) | `vector`

Mechanical properties of the motor:

Inertia, J, in kg.m^2
Viscous damping, F, in N·m/(rad/s)
Static friction, T_f, in N·m

Dependencies

To enable this parameter, set the Mechanical input configuration parameter to Torque.

Initial Values

Initial d-axis and q-axis current (idq0) — Current
`[0 0]` (default) | `vector`

Initial q- and d-axis currents, i_q, i_d, in A.

Initial mechanical position (theta_init) — Angle
`0` (default) | `scalar`

Initial motor angular position, θ_m0, in rad.

Initial mechanical speed (omega_init) — Speed
`0` (default) | `scalar`

Initial angular velocity of the motor, ω_m0, in rad/s.

Dependencies

To enable this parameter, set the Mechanical input configuration parameter to Torque.

References

[1] Kundur, P. Power System Stability and Control. New York, NY: McGraw Hill, 1993.

[2] Anderson, P. M. Analysis of Faulted Power Systems. Hoboken, NJ: Wiley-IEEE Press, 1995.

Extended Capabilities

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

Version History

Introduced in R2017a

Surface Mount PMSM

Description

Motor Construction

Three-Phase Sinusoidal Model Electrical System

Mechanical System

Power Accounting

Amplitude Invariant dq Transformation

Ports

Input

LdTrq — Load torque on motor scalar

Dependencies

Spd — Motor shaft speed scalar

Dependencies

PhaseVolt — Stator terminal voltages 1-by-3 array

Output

Info — Bus signal bus

PhaseCurr — Phase a, b, c current 1-by-3 array

MtrTrq — Motor torque scalar

Dependencies

MtrSpd — Motor speed scalar

Dependencies

Parameters

Mechanical input configuration — Select port configuration Torque (default) | Speed

Simulation type — Select simulation type Continuous (default) | Discrete

Dependencies

Sample Time (Ts) — Sample time for discrete integration 0.001 (default) | scalar

Dependencies

Number of pole pairs (P) — Pole pairs 4 (default) | scalar

Stator resistance per phase (Rs) — Resistance .2 (default) | scalar

Stator d-axis inductance (Ldq_) — Inductance 3.752e-4 (default) | scalar

Permanent flux linkage constant (lambda_pm) — Flux 0.1194 (default) | scalar

Back-emf constant (Ke) — Back electromotive force scalar

Torque constant (Kt) — Torque constant scalar

Physical inertia, viscous damping, static friction (mechanical) — Inertia, damping, friction [0.002700,4.924e-4,0] (default) | vector

Dependencies

Initial d-axis and q-axis current (idq0) — Current [0 0] (default) | vector

Initial mechanical position (theta_init) — Angle 0 (default) | scalar

Initial mechanical speed (omega_init) — Speed 0 (default) | scalar

Dependencies

References

Extended Capabilities

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

Version History

See Also

Topics

LdTrq — Load torque on motor
`scalar`

Spd — Motor shaft speed
`scalar`

PhaseVolt — Stator terminal voltages
`1`-by-`3` array

Info — Bus signal
bus

PhaseCurr — Phase a, b, c current
`1`-by-`3` array

MtrTrq — Motor torque
`scalar`

MtrSpd — Motor speed
`scalar`

Mechanical input configuration — Select port configuration
`Torque` (default) | `Speed`

Simulation type — Select simulation type
`Continuous` (default) | `Discrete`

Sample Time (Ts) — Sample time for discrete integration
`0.001` (default) | `scalar`

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

Stator resistance per phase (Rs) — Resistance
`.2` (default) | `scalar`

Stator d-axis inductance (Ldq_) — Inductance
`3.752e-4` (default) | `scalar`

Permanent flux linkage constant (lambda_pm) — Flux
`0.1194` (default) | `scalar`

Back-emf constant (Ke) — Back electromotive force
`scalar`

Torque constant (Kt) — Torque constant
`scalar`

Physical inertia, viscous damping, static friction (mechanical) — Inertia, damping, friction
`[0.002700,4.924e-4,0]` (default) | `vector`

Initial d-axis and q-axis current (idq0) — Current
`[0 0]` (default) | `vector`

Initial mechanical position (theta_init) — Angle
`0` (default) | `scalar`

Initial mechanical speed (omega_init) — Speed
`0` (default) | `scalar`

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