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# PMSM (Single-Phase)

Single-phase permanent magnet synchronous motor

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• Simscape / Electrical / Electromechanical / Permanent Magnet

## Description

The PMSM (Single-Phase) represents a single-phase permanent magnet synchronous motor (PMSM), a type of DC motor that is useful for automation applications.

The figure shows the topology of the single-phase PMSM drive.

The figure shows the motor construction with a single pole-pair on the rotor. Single-phase PMSMs are not self-starting unless the air gap is asymmetrical.

The figure shows the equivalent circuit for the PMSM (Single-Phase) block.

### Equations

The motor voltage equations are

`${v}_{s}=Ri+L\frac{di}{dt}+e$`

and

`${v}_{s}={V}_{m}\text{sin}\left({\omega }_{s}t+\epsilon \right),$`

where:

• vs is the supply voltage.

• i is the instantaneous motor current.

• R is the resistance of the windings.

• L is the self-inductance of the windings.

• e is the back-electromotive force (BEMF).

• ɷs is the angular frequency of the supply voltage.

• ε is the angle of the supply voltage.

The back electro-motive force (BEMF) is

`$e={k}_{e}{\omega }_{r}\text{sin}\left({\theta }_{r}\right),$`

where:

• ɷr is the rotor electrical angular velocity.

• θr is the rotor position.

• ke is the BEMF constant.

Due to the large low-permeability gaps between the stator and rotor, the saturation can be neglected. Therefore, the electric torque equations are

`${T}_{e}=i{\psi }_{m}\text{sin}\left({\theta }_{r}\right)$`

and

`${\psi }_{m}=\frac{{k}_{e}}{p},$`

where:

• Te is the electric torque.

• ψm is the permanent magnet flux linkage.

• p is the number of pole pairs.

The mechanical equation is

`${J}_{m}\frac{d{\omega }_{r}}{dt}={T}_{e}-{T}_{L}-{B}_{m}{\omega }_{r},$`

where:

• Jm is the rotor inertia.

• TL is the torque load.

• Bm is the friction coefficient.

### Variables

Use the Variables settings to specify the priority and initial target values for the block variables before simulation. For more information, see Set Priority and Initial Target for Block Variables (Simscape).

## Limitations and Assumptions

• The machine air gap is free of saliency effects.

• The stator current has negligible effect on the flux distribution under normal operating conditions.

• The hysteresis, saturation effects, and eddy currents are neglected.

## Ports

### Conserving

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Mechanical rotational conserving port associated with the machine rotor.

Mechanical rotational conserving port associated with the machine case.

Electrical conserving port associated with the supply positive terminal.

Electrical conserving port associated with the supply negative terminal.

## Parameters

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### Main

Number of permanent magnet pole pairs on the rotor.

Method for parameterizing the stator.

#### Dependencies

Selecting `Specify flux linkage` exposes the Permanent magnet flux linkage parameter.

Selecting `Specify back EMF constant` exposes the Back EMF constant parameter.

#### Dependencies

Selecting `Specify flux linkage` for the Permanent magnet flux linkage parameterization parameter exposes the Permanent magnet flux linkage parameter.

Back-electromotive force constant.

#### Dependencies

Selecting `Specify back EMF constant` for the Permanent magnet flux linkage parameterization parameter exposes the Back EMF constant parameter.

The direct-axis inductance.

Resistance of each of the stator windings.

Rotor angular position at standstill due to asymmetric air gap.

### Mechanical

Inertia of the rotor attached to mechanical translational port R.

Damping of the rotor.

## References

[1] Ertugrul, N. and C. Doudle. “Dynamic analysis of a single-phase line-starting permanent magnet synchronous motor.” Proceedings of International Conference on Power Electronics, Drives and Energy Systems for Industrial Growth. Vol. 1, 1996, pp. 603–609.