Electric Machines

The model for PMSMs is based on that in Dowdle et al.^[1], with some modifications for increased fidelity. The PMSM consists of:

• A rotor back iron connected to a shaft.
• A magnet with a number of pole pairs (p) attached to the rotor back iron.
• A thin air gap separating the rotor from the stator.
• A stator consisting of metallic teeth supported by a back iron. Slots between the teeth contain Litz wires.

This implementation supports an arbitrary number of electrical phases, although the source model was designed for three-phase power.

PMSM sizing

The following parameters are required for PMSM sizing:

• Number of pole pairs ($p$).
• Magnet thickness ($t_M$).
• Air gap thickness ($t_\mathrm{gap}$).
• Maximum rotor linear velocity ($U_\mathrm{max}$).
• Ratio of metal flux density to saturation flux density ($r_\mathrm{sat}$).
• Number of slots per pole ($N_{sp}$).
• Teeth thickness ($t_\mathrm{teeth}$).
• Shaft speed ($\Omega$).
• Shaft power ($P_\mathrm{shaft}$).
• Maximum wire current density ($j_\mathrm{max}$).
• Slot packing factor ($k_{pf}$)
• Component materials.

The radius at the gap start $R_\mathrm{gap}$ is calculated from the maximum rotational speed as

\[ R_\mathrm{gap} = \frac{U_\mathrm{max}}{\Omega}.\]

The air gap magnetic field flux density, $B_\mathrm{gap}$ can be computed from the magnet properties as^[1]

\[ B_\mathrm{gap} = \frac{\mu_0 M t_M}{t_M + t_\mathrm{gap}},\]

where $\mu_0$ is the vacuum permeability and $M$ is the magnet's magnetization constant. The magnetic flux densities in the rotor and stator back irons can be computed by

\[ B_x = r_\mathrm{sat} B_\mathrm{sat},\]

where $x$ refers to the rotor or stator back irons and $B_\mathrm{sat}$ is the back iron material's saturation flux density. The stator and rotor back iron thicknesses can then be computed by

\[ t_x = \frac{B_\mathrm{gap} \pi R_\mathrm{gap}}{B_x 2 p}.\]

From the geometry, the rotor back iron outer radius is simply $R_{r,o}=R_\mathrm{gap}-t_M$ and its inner radius is $R_{r,i}=R_{r,o}-t_{r}$, where $t_{r}$ is the rotor back iron thickness.

The magnetic flux density produced by the windings on the teeth is given by^[1]

\[ B_\mathrm{wind} = \mu_0 j_\mathrm{max} k_{pf} t_\mathrm{teeth},\]

where $k_{pf}$ is the slot packing factor. This adds vectorially to the teeth flux due to the permanent magnets,

\[ B_\mathrm{teeth}^2 = B_\mathrm{wind}^2 + \left(B_\mathrm{gap}\frac{A_\mathrm{ann}}{A_\mathrm{slots}}\right)^2,\]

where $A_\mathrm{ann} = \pi\left((R_{s,i}+t_\mathrm{teeth})^2-R_{s,i}^2\right)$ is the area of the annulus where the teeth and slots are located and $A_\mathrm{slots}$ is the total area of the slots. If the flux density on the teeth is given by $B_\mathrm{teeth} = r_\mathrm{sat} B_\mathrm{sat}$, the slot total area is

\[ A_\mathrm{teeth} = A_\mathrm{ann} - A_\mathrm{ann}\frac{B_\mathrm{gap}}{\sqrt{B_\mathrm{teeth}^2 - B_\mathrm{wind}^2}} \]

Since the number of teeth is equal to the number of slots and is $N_\mathrm{teeth}=2pN_{sp}$, the width of a single tooth is $w_\mathrm{tooth} = \frac{A_\mathrm{teeth}}{t_\mathrm{teeth} N_\mathrm{teeth}}$. The area of a single slot, $A_\mathrm{slot}$ is given by

\[ A_\mathrm{slot} = \frac{A_\mathrm{ann} - A_\mathrm{teeth}}{N_\mathrm{teeth}}. \]

The current in a slot, $I$, is given by

\[ I = j_\mathrm{max} A_\mathrm{slot} k_{pf}. \]

The length of the PMSM, $l$, can be computed from knowledge of the torque $T=P_\mathrm{shaft}/\Omega$,

\[ l = \frac{T}{R_\mathrm{gap} I B_\mathrm{gap} N_{es}}, \]

where $N_{es}$ is the number of energized slots. The masses of the different components can then be estimated from their respective volumes and densities.

PMSM operation

Motor Operation

For motors, the input power is calculated as:

\[P_\mathrm{input} = P_\mathrm{shaft} + \dot{Q}_\mathrm{loss},\]

where $\dot{Q}_\mathrm{loss}$ includes all relevant power losses.

Generator Operation

For generators, the output power is calculated as:

\[P_\mathrm{output} = P_\mathrm{shaft} - \dot{Q}_\mathrm{loss}.\]

Loss Calculations

The total power loss in a PMSM is the sum of ohmic, core, and windage losses,

\[\dot{Q}_\mathrm{loss} = \dot{Q}_\mathrm{ohmic} + \dot{Q}_\mathrm{core} + \dot{Q}_\mathrm{wind}.\]

Ohmic Losses:

\[Q_\mathrm{ohmic} = I^2 R_\mathrm{phase} N_\mathrm{ep},\]

where $I$ is the current, $R_\mathrm{phase}$ is the phase resistance, and $N_\mathrm{ep}$ is the number of energized phases.

Core Losses: Core losses include hysteresis and eddy current losses:

\[\dot{Q}_\mathrm{core} = \dot{Q}_\mathrm{hysteresis} + \dot{Q}_\mathrm{eddy}.\]

Hysteresis Losses:

\[\dot{Q}_\mathrm{hysteresis} = k_h f B^α,\]

where $k_h$ is the hysteresis coefficient, $f$ is the frequency, $B$ is the flux density, and $α$ is a material constant.

Eddy Current Losses:

\[\dot{Q}_\mathrm{eddy} = k_e f^2 B^2,\]

where $k_e$ is the eddy current coefficient.

Windage Losses: Windage losses are caused by air friction and are calculated using Vrancik's model^[2],

\[\dot{Q}_\mathrm{wind} = C_f \pi \rho \Omega^3 R_\mathrm{gap}^4 l,\]

where $C_f$ is the skin friction coefficient, $\rho$ is the air density, $\Omega$ is the angular velocity, $R_\mathrm{gap}$ is the gap radius, and $l$ is the stack length.

Back e.m.f.

For motors, the voltage $V$ corresponding to the power demand of the motor is calculated as

\[ V = \frac{P_\mathrm{input}}{N_\mathrm{phases} \frac{2}{\pi} I N_\mathrm{inverters}},\]

where $N_\mathrm{phases}$ is the number of phases and $N_\mathrm{inverters}$ is the number of inverters used to power the motor. A similar expression is used to calculate the output voltage of the generator, without the number of inverters.