Permanent magnet synchronous motor with sinusoidal flux distribution

**Library:**Simscape / Electrical / Electromechanical / Permanent Magnet

The PMSM block models a permanent magnet synchronous motor with a three-phase wye-wound stator. The figure shows the equivalent electrical circuit for the stator windings.

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

Permanent magnets generate a rotor magnetic field that creates a sinusoidal rate of change of flux with rotor angle.

For the axes convention in the preceding figure, the *a*-phase
and permanent magnet fluxes are aligned when rotor mechanical angle,
*θ _{r}*, is zero. The block supports a
second rotor axis definition in which rotor mechanical angle is defined as the angle
between the

Voltages across the stator windings are defined by:

$$\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]=\left[\begin{array}{ccc}{R}_{s}& 0& 0\\ 0& {R}_{s}& 0\\ 0& 0& {R}_{s}\end{array}\right]\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]+\left[\begin{array}{c}\frac{d{\psi}_{a}}{dt}\\ \frac{d{\psi}_{b}}{dt}\\ \frac{d{\psi}_{c}}{dt}\end{array}\right],$$

where:

*v*,_{a}*v*, and_{b}*v*are the individual phase voltages across the stator windings._{c}*R*is the equivalent resistance of each stator winding._{s}*i*,_{a}*i*, and_{b}*i*are the currents flowing in the stator windings._{c}$$\frac{d{\psi}_{a}}{dt},$$$$\frac{d{\psi}_{b}}{dt},$$ and $$\frac{d{\psi}_{c}}{dt}$$ are the rates of change of magnetic flux in each stator winding.

The permanent magnet and the three windings contribute to the total flux linking each winding. The total flux is defined by:

$$\left[\begin{array}{c}{\psi}_{a}\\ {\psi}_{b}\\ {\psi}_{c}\end{array}\right]=\left[\begin{array}{ccc}{L}_{aa}& {L}_{ab}& {L}_{ac}\\ {L}_{ba}& {L}_{bb}& {L}_{bc}\\ {L}_{ca}& {L}_{cb}& {L}_{cc}\end{array}\right]\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]+\left[\begin{array}{c}{\psi}_{am}\\ {\psi}_{bm}\\ {\psi}_{cm}\end{array}\right],$$

where:

*ψ*,_{a}*ψ*, and_{b}*ψ*are the total fluxes linking each stator winding._{c}*L*,_{aa}*L*, and_{bb}*L*are the self-inductances of the stator windings._{cc}*L*,_{ab}*L*,_{ac}*L*, and so on, are the mutual inductances of the stator windings._{ba}*ψ*,_{am}*ψ*, and_{bm}*ψ*are the permanent magnet fluxes linking the stator windings._{cm}

The inductances in the stator windings are functions of rotor electrical angle, defined by:

${\theta}_{e}=N{\theta}_{r},$

${L}_{aa}={L}_{s}+{L}_{m}\text{cos}(2{\theta}_{e}),$

${L}_{bb}={L}_{s}+{L}_{m}\text{cos}(2\left({\theta}_{e}-2\pi /3\right)),$

${L}_{cc}={L}_{s}+{L}_{m}\text{cos}(2\left({\theta}_{e}+2\pi /3\right)),$

$${L}_{ab}={L}_{ba}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{e}+\pi /6\right)\right),$$

${L}_{bc}={L}_{cb}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{e}+\pi /6-2\pi /3\right)\right),$

and

${L}_{ca}={L}_{ac}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{e}+\pi /6+2\pi /3\right)\right),$

where:

*θ*is the rotor mechanical angle._{r}*θ*is the rotor electrical angle._{e}*L*is the stator self-inductance per phase. This value is the average self-inductance of each of the stator windings._{s}*L*is the stator inductance fluctuation. This value is the amplitude of the fluctuation in self-inductance and mutual inductance with changing rotor angle._{m}*M*is the stator mutual inductance. This value is the average mutual inductance between the stator windings._{s}

The permanent magnet flux linking winding *a* is a maximum when
*θ _{e}* = 0° and zero when

$$\left[\begin{array}{c}{\psi}_{am}\\ {\psi}_{bm}\\ {\psi}_{cm}\end{array}\right]=\left[\begin{array}{c}{\psi}_{m}\mathrm{cos}{\theta}_{e}\\ {\psi}_{m}\mathrm{cos}\left({\theta}_{e}-2\pi /3\right)\\ {\psi}_{m}\mathrm{cos}\left({\theta}_{e}+2\pi /3\right)\end{array}\right].$$

where *ψ _{m}* is the permanent magnet flux
linkage.

Applying Park’s transformation to the block electrical equations produces an expression for torque that is independent of the rotor angle.

Park’s transformation is defined by:

$P=2/3\left[\begin{array}{ccc}\mathrm{cos}{\theta}_{e}& \mathrm{cos}\left({\theta}_{e}-2\pi /3\right)& \mathrm{cos}\left({\theta}_{e}+2\pi /3\right)\\ -\mathrm{sin}{\theta}_{e}& -\mathrm{sin}\left({\theta}_{e}-2\pi /3\right)& -\mathrm{sin}\left({\theta}_{e}+2\pi /3\right)\\ 0.5& 0.5& 0.5\end{array}\right].$

where *θ _{e}* is the electrical angle defined
as

Using Park's transformation on the stator winding voltages and currents transforms them to the dq0 frame, which is independent of the rotor angle:

$$\left[\begin{array}{c}{v}_{d}\\ {v}_{q}\\ {v}_{0}\end{array}\right]=P\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]$$

and

$\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]=P\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right].$

Applying Park’s transformation to the first two electrical equations produces the following equations that define the block behavior:

${v}_{d}={R}_{s}{i}_{d}+{L}_{d}\frac{d{i}_{d}}{dt}-N\omega {i}_{q}{L}_{q},$

${v}_{q}={R}_{s}{i}_{q}+{L}_{q}\frac{d{i}_{q}}{dt}+N\omega ({i}_{d}{L}_{d}+{\psi}_{m}),$

${v}_{0}={R}_{s}{i}_{0}+{L}_{0}\frac{d{i}_{0}}{dt},$

and

$T=\frac{3}{2}N\left({i}_{q}\left({i}_{d}{L}_{d}+{\psi}_{m}\right)-{i}_{d}{i}_{q}{L}_{q}\right),$

where:

*L*=_{d}*L*+_{s}*M*+ 3/2_{s}*L*._{m}*L*is the stator_{d}*d*-axis inductance.*L*=_{q}*L*+_{s}*M*− 3/2_{s}*L*._{m}*L*is the stator_{q}*q*-axis inductance.*L*=_{0}*L*– 2_{s}*M*._{s}*L*is the stator zero-sequence inductance._{0}*ω*is the rotor mechanical rotational speed.*N*is the number of rotor permanent magnet pole pairs.*T*is the rotor torque. Torque flows from the motor case (block physical port C) to the motor rotor (block physical port R).

The PMSM block uses the original, non-orthogonal implementation of the Park transform. If you try to apply the alternative implementation, you get different results for the dq0 voltage and currents.

You can parameterize the motor using the back EMF or torque constants which are
more commonly given on motor datasheets by using the **Permanent magnet flux
linkage** option.

The back EMF constant is defined as the peak voltage induced by the permanent magnet in each of the phases per unit rotational speed. It is related to peak permanent magnet flux linkage by:

$${k}_{e}=N{\psi}_{m}.$$

From this definition, it follows that the back EMF
*e _{ph}* for one phase is given
by:

$${e}_{ph}={k}_{e}\omega .$$

The torque constant is defined as the peak torque induced by each of the phases per unit current. It is numerically identical in value to the back EMF constant when both are expressed in SI units:

$${k}_{t}=N{\psi}_{m}.$$

When
*L _{d}*=

$$T=\frac{3}{2}{k}_{t}{i}_{q}=\frac{3}{2}{k}_{t}{I}_{pk},$$

where *I _{pk}* is the peak current in any of
the three windings.

The factor 3/2 follows from this being the steady-state sum of the torques from
all phases. Therefore the torque constant
*k _{t}* could also be defined as:

$${k}_{t}=\frac{2}{3}\left(\frac{T}{{I}_{pk}}\right),$$

where *T* is the measured total torque when
testing with a balanced three-phase current with peak line voltage
*I _{pk}*. Writing in terms of RMS line voltage:

$${k}_{t}=\sqrt{\frac{2}{3}}\left(\frac{T}{{i}_{line,rms}}\right).$$

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).

[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.