I have been working on a dynamic model of the PMSM with surface mounted magnets and sinusoidal stator winding. I am little bit confused by formulas for the \$L_d\$ and \$L_q\$ inductances. I have found several formulas for these inductances and I don't know which of them are correct.

So I have attempted to derive the equations alone and I have followed these steps

1. Formulas for magnetic fluxes for individual stator phases (\$L\$ stator phase self inductance, \$M\$ mutual inductance for stator windings, \$\psi_m\$ magnetic flux of permanent magnet,\$\theta\$ rotor position i.e. angle between axes of stator phase \$a\$ and permanent magnet axes \$d\$) enter image description here

2. Clarke transform (\$\hat{a}=e^{j\frac{2\pi}{3}}\$)

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3. Magnetic fluxes in \$(\alpha,\beta)\$

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4. Park transform

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5. Magnetic fluxes in \$(d,q)\$

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Based on the formulas for magnetic fluxes in \$(d,q)\$ I have following formulas \$L_d=\frac{2}{3}(L-M)\$ and \$L_q=\frac{2}{3}(L-M)\$. Is my derivation correct? I am confused because my formulas are totally different from what I have seen so far. Thank you for any ideas.


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