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Air gap between stator tooth and rotating permanent magnet rotor

**Library:**Simscape / Electrical / Electromechanical

The Rotating Air Gap block models an air gap between a stator tooth and a rotating permanent magnet rotor. This block assumes that the rotor magnets are surface mounted and that the associated induced voltage is sinusoidal.

If the rotor angle is zero, specified by the **Rotor angle** variable
in the **Variables** tab, then the rotor magnet perfectly aligns with
the middle of the first stator tooth. The permanent magnet is then orientated to oppose
the flux flow from port **N** to port **S**.

Use this block to create a magnetic representation of a permanent magnet synchronous motor
(PMSM). For example, if you want to model a motor with nine stator poles, create nine
copies of this block and set each of the **Stator tooth reference
index** parameters to `1`

, `2`

,
`3`

, `4`

, `5`

,
`6`

, `7`

, `8`

, and
`9`

, respectively.

This figure shows the equivalent circuit for the air gap and the adjacent permanent magnet

where:

*ϕ*is the magnetic flux that flows from the external magnetic circuit to port_{g}**N**.*R*is the air gap reluctance._{g}*mmf*is the magnetomotive force across the rotating air gap component.*R*is the permanent magnet reluctance._{m}*ϕ*is the magnetic flux generated by the rotor permanent magnets in the angle range subtended by the stator tooth._{r}

This equation defines the relationship between
*ϕ _{g}*,

$${\varphi}_{g}=\frac{mmf-{R}_{m}{\varphi}_{r}}{{R}_{m}+{R}_{g}}.$$

If the back EMF is sinusoidal, the flux density of the permanent magnet rotor is defined by this equation

$${B}_{r}={B}_{0}cos\left(N{\theta}_{s}-N{\theta}_{r}\right)$$

where:

*N*is the**Number of rotor pole pairs**.*θ*is the rotor angle._{r}*θ*is the stator angle._{s}*B*is the_{0}**Peak magnet flux density**, in Tesla.

Then, to obtain the permanent magnet flux linkage, integrate over the stator angle subtended by the stator tooth

$${\varphi}_{r}({\theta}_{r})=rl{\displaystyle {\int}_{\frac{-{\theta}_{tooth}}{2}}^{\frac{{\theta}_{tooth}}{2}}\left[{B}_{0}\mathrm{cos}\left(N{\theta}_{s}-N{\theta}_{r}\right)\right]d{\theta}_{s}}$$

where:

*r*is the**Rotor radius**.*l*is the**Tooth depth (in direction of shaft)**.

For an ideal PMSM, the *θ _{tooth}* must be equal to

$${\varphi}_{r}({\theta}_{r})=2{B}_{0}lr/Nsin\left(\frac{\pi N}{{N}_{s}}\right)\mathrm{cos}\left(N{\theta}_{r}\right).$$

To obtain the torque generated across the air gap, first calculate the total energy stored by the component:

$$E=\frac{1}{2}{\varphi}_{g}^{2}{R}_{g}+\frac{1}{2}{\left({\varphi}_{r}\left({\theta}_{r}\right)\right)}^{2}{R}_{m}.$$

Then, to obtain the torque, differentiate with respect to the rotor angle:

$$\tau =\frac{\partial E}{\partial {\theta}_{r}}=-2{B}_{0}{R}_{m}lrsin\left(\frac{\pi N}{{N}_{s}}\right)\mathrm{sin}\left(N{\theta}_{r}\right)\left({\varphi}_{g}+{\varphi}_{r}\left({\theta}_{r}\right)\right)/N.$$

Finally, calculate *R _{g}* and

$$\begin{array}{l}{R}_{g}=\frac{g}{{\mu}_{0}{A}_{g}}\\ {R}_{m}=\frac{{l}_{m}}{{\mu}_{r}{\mu}_{0}{A}_{g}}\end{array}$$

where:

*μ*is the permittivity of free space._{0}*μ*is the relative permittivity of the permanent magnet._{r}*g*is the**Air gap**.*l*is the magnet length._{m}

Use the **Variables** section of the block
interface to set the priority and initial target values for the block
variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables.