Surface-mounted permanent magnet synchronous motor computation

This model represent a SM PMSM with non-salient radial flux and supplied with sinewave currents. This choice simplifies the modeling process while providing accurate representation of the PMSM characteristic. As in other PMSMs, the stator bore slots between adjacent teeth hold the wound cables, serving as the electromagnets. Unlike AF PMSM, the permanent magnets are mounted directly on the rotor’s outer surface rather than embedded within it.

../../../../../_images/cylindrical_pmsm.svg

This figure, provided by one of the main thesis of HASTECS project [Tou20], demonstrates the essential geometric parameters of a SM PMSM. \(L_m\) is the length of the motor’s active electromagnetic part.

Performance calculation

Most of the performance calculations shares a similar approach as the AF PMSM model, except the losses caused by various factors. The three primary sources of losses in the Surface-mounted PMSM as modeled in the HASTECS project [Tou20] are presented below.

PMSM Joule losses

Joule losses represent the most important sources of losses in the SMPMSM. It is caused by Ohmic heating in the conductor wires winding of the PMSM stator. At low operating speeds, the current density is evenly distributed across the wire’s cross-section. To calculate the Joule losses, the wire resistance must be determined first.

\[\begin{split}R_s = \frac{N_c}{q} \cdot \rho_{cu}(T_{win}) \\ \rho_{cu}(T_{win}) = \rho_{cu20^\circ} [1 + \alpha_{th}(T_{win} - 20^\circ)]\end{split}\]

Where \(N_c\) is the number of conductors, \(q\) is the number of phase in PMSM, \(T_{win}\) is the the temperature of the wire windings, \(\alpha_{th}\) is the electrical resistance coefficient of copper, and \(\rho_{cu20^\circ}\) copper density at \(20^{\circ}C\).

With the wire electrical resistance and the RMS current (\(I_{rms}\)), the Joule losses can be written as:

\[P_j = q \cdot R_s \cdot I_{rms}^2\]

PMSM iron losses

As the second largest contributor to the PMSM performance losses, the iron losses arises from eddy current and the continuous variation of the magnetic flux. To better capture the behavior of the SM PMSM, a regression model using Least Squared Method by HASTECS project [Tou20] is considered.

\[P_{ir} = \sum_{i=1}^{i=4}\sum_{j=1}^{j=4} a_{ij}(\sqrt{B_m})^j(\sqrt{f})^i\]

\(B_m\) is the maximum magnetic flux density and \(f\) is the magnetic field switching frequency.

../../../../../_images/iron_losses.svg

The iron losses coefficients (\(a_{ij}\)) are verified with empirical data provided by HASTECS project [Tou20].

PMSM mechanical losses

Mechanical losses (\(P_{mech}\)) are the consequence of various phenomenons including, friction between air and rotor or friction between a stationary solid and a rotating solid.

\[P_{mech} = P_{windage} + 2 P_{bf}\]

The two major windage losses (\(P_{windage}\)) result from the fluid friction between the air in the component gaps and the rotor. The airgap windage losses (\(P_{wa}\)) occurs because of the fluid friction between the stator and rotor while rotating. Similarly, the rotor windage losses (\(P_{wr}\)) arises from the space between both ends of the rotor and the motor casing. The rotor radius is denoted as \(R_{r}\), the shaft radius as \(R_{sh}\), the rotation speed as \(\Omega\), the air density as \(\rho_{air}\), and the motor length as \(L\).

\[\begin{split}P_{windage} = P_{wa} + 2P_{wr} \\ P_{wa} = k_1 C_{fa} \pi \rho_{air} \Omega^3 R_r^4 L \\ P_{wr} = \frac{1}{2}C_{fr} \pi \rho_{air} \Omega^3(R_r^5 - R_{sh}^5)\end{split}\]

Where the friction coefficient of airgap windage losses (\(C_{fa}\)) and the friction coefficient of rotor windage losses (\(C_{fr}\)) are:

\[\begin{split}C_{fa} = \begin{cases} 0.515 \frac{(e_g/R_r)^{0.3}}{Re_{a}^{0.5}} & \text{for laminar flow } 500 < Re_{a} < 10^4 \\ 0.0325 \frac{(e_g/R_r)^{0.3}}{Re_{a}^{0.2}} & \text{for turbulent flow } Re_{a} > 10^4 \end{cases} \\\end{split}\]
\[\begin{split}C_{fr} = \begin{cases} \frac{3.87}{Re_{rt}^{0.5}} & \text{for laminar flow } Re_{rt} \leq 3.5 \cdot 10^5 \\ \frac{0.146}{Re_{rt}^{0.2}} & \text{for turbulent flow } Re_{rt} > 3.5 \cdot 10^5 \end{cases} \\\end{split}\]

\(e_g\) is the airgap thickness.

With the air pressure expressed as \(pr\), the air density (\(\rho_{air}\)) and the air dynamic viscosity (\(\mu_{air}\)).

And the Reynolds numbers for both losses are:

\[\begin{split}Re_{a} = \frac{\rho_{air} R_r e_g}{\Omega} \\ Re_{rt} = \frac{\rho_{air} R_r^2}{\mu_{air}} \Omega\end{split}\]

The bearing friction losses is another major contributor to the friction losses between a moving surface and a stationary surface. A simplified model for the bearing friction coefficient (\(C_{fb}\)) is provided for various bearing type based on SKF’s bearing datasheets [SKFGroup16].

Bearing types

Friction coefficient \(C_{fb}\)

Deep groove ball bearings

\(0.0015\)

Cylindrical roller bearings

  • with cage

\(0.0011\)

  • full complement

\(0.0020\)

Spherical toroidal roller bearings

\(0.0018\)

CARB toroidal roller bearings

\(0.0016\)

Angular contact ball bearings

  • single row

\(0.0020\)

  • double row

\(0.0024\)

  • four-point contact

\(0.0024\)

Hybrid bearings

–
\[\begin{split}P_{bf} = \frac{1}{2}C_{fb} \cdot P \cdot d_{bb} \cdot \Omega \\ P = W_{rt} \cdot g\end{split}\]

\(W_{rt}\) is the rotor weight, \(d_{bb}\) is the bearing bore diameter and the \(g\) is the gravitational constant.

Sizing calculation

In this sizing process, several geometry parameters related to the electromagnetic parts of the PMSM are simplified for a simpler model. The rotor is modeled as a single solid rod, omitting the bore layer and surface magnet sheets, with its material density defined according to the Etel TMB and TMK electric motor dataset from the HASTECS project [Tou20]. The slot geometry is modeled as a rectangle with no radial taper and without fillets.

SM PMSM dimension calculation

From the electric current balance and magnetic flux balance, the stator bore radius (\(R_{rt}\)), the active length (\(L_{m}\)), the conductor slot height (\(h_{s}\)), and the yoke thickness (\(h_{y}\)) can be derived.

\[\begin{split}R_{rt} = \sqrt[3]{\frac{\lambda}{4\pi\sigma}\frac{P_{em}}{\Omega}} \\ L_m = (\frac{2}{\lambda})\sqrt[3]{\frac{\lambda}{4\pi\sigma}\frac{P_{em}}{\Omega}}\end{split}\]

\(\lambda = 2 R/L_m\) is the shape coefficient, \(\sigma\) is the tangential stress, and \(P_{em}\) is the given electromagnetic power.

\[h_s = \frac{\sqrt{2}\sigma}{k_w B_m j_{rms} k_{sc} k_{fill}} (1-r_{tooth})^{-1}\]
\[h_y = \frac{R_{rt}}{p} \sqrt{(\frac{B_{m}}{B_{sy}})^2 + \mu_o^2 (\frac{K_m}{B_{sy}})^2 \tau_{x2p}^2}\]
\[\begin{split}r_{tooth} = \frac{2}{\pi} \sqrt{(\frac{B_{m}}{B_{st}})^2 + \mu_o^2 (\frac{K_m}{B_{st}})^2 \tau_{x2p}^2} \\ \tau_{x2p}^2 = \frac{1+x^{2p}}{1-x^{2p}}\end{split}\]

Variable

Explanation

\(B_m\)

Max airgap magnetic flux density

\(K_m\)

Max electric surface current density

\(B_{st}\)

Magnetic flux density in teeth

\(B_{sy}\)

Magnetic flux density in the yoke

\(j_{rms}\)

RMS current density

\(p\)

Number of pole pairs

\(k_{fill}\)

Cross section ratio between a slot and the wires in the slots

\(k_{sc}\)

Wire cross section ratio between straight cut and tilted cut

\(k_w\)

Wire winding coefficient

\(x\)

Radius ratio of the rotor radius and the stator bore radius

SM PMSM weight calculation

The weight of the SM PMSM is the sum of the weights of all fundamental components, the stator core weight (\(W_{stc}\)), the stator winding weight (\(W_{stw}\)), the rotor weight (\(W_{rt}\)) , and the frame weight (\(W_{f}\)).

\[W_{stc} = [\pi \cdot L_m (R_{out}^2-R^2) - (h_s \cdot L_m \cdot N_s \cdot l_s)] \rho_{stc}\]
\[W_{stw} = [k_{tb} k_{tc} h_s L_m N_s l_s][k_{fill} \rho_c (1 - k_{fill}) \rho_{ins}]\]

Variable

Explanation

\(N_s\)

Number of the wire slots

\(k_{tb}\)

Cross section ratio between a slot and the wires in the slots

\(k_{tc}\)

Conductor wire twisting coefficient

\(ls\)

Slot width

\(\rho_{stc}\)

Stator core material density

\(\rho_{stw}\)

Stator winding (teeth) material density

\(\rho_{c}\)

Conductor wire material density

\(\rho_{ins}\)

Wire insulation material density
\[\begin{split}W_{rt} = \pi R_r^2 L_m \rho_{rt}(p) \\ \rho_{rt}(p) = \begin{cases} −431.67 p + 7932 & \text{for} p \leq 10 \\ 1.09 p^2 − 117.45 p + 4681 & \text{for} 10 < p \leq 50 \\ 1600 & \text{for} p > 50 \end{cases} \\\end{split}\]

\(R_r\) is the rotor radius and the \(\rho_{rt}\) is the rotor material density.

\[\begin{split}W_{f} = \rho_{fr} (\pi L_m k_{tb} (R_{fr}^2 - R_{out}^2) + 2 \pi (\tau_r(R_{out}) - 1) R_{out} R_{fr}^2) \\ \tau_r(R_{out}) = \begin{cases} 0.7371 R_{out}^2 − 0.580 R_{out} + 1.1599 & \text{for} R_{out} \leq 400mm \\ 1.04 & \text{for} R_{out} > 400mm \\ \end{cases} \\\end{split}\]

\(R_{fr}\) is the frame radius, \(R_{out}\) is the outer stator diameter, and \(\tau_r\) is the ratio of \(R_{fr}\) and \(R_{out}\).

Component Computation Structure

The following two links are the N2 diagrams representing the performance and sizing computation for the Surface-Mounted PMSM model.

SM PMSM performance N2 diagram
SM PMSM sizing N2 diagram