PMSM dynamic model - Electrical Engineering Stack Exchange most recent 30 from electronics.stackexchange.com 2019-09-19T17:48:03Z https://electronics.stackexchange.com/feeds/question/417531 https://creativecommons.org/licenses/by-sa/4.0/rdf https://electronics.stackexchange.com/q/417531 1 PMSM dynamic model Steve https://electronics.stackexchange.com/users/69769 2019-01-17T20:23:52Z 2019-01-17T20:23:52Z <p>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 <span class="math-container">\$L_d\$</span> and <span class="math-container">\$L_q\$</span> inductances. I have found several formulas for these inductances and I don't know which of them are correct. </p> <p>So I have attempted to derive the equations alone and I have followed these steps</p> <p><strong>1. Formulas for magnetic fluxes for individual stator phases</strong> (<span class="math-container">\$L\$</span> stator phase self inductance, <span class="math-container">\$M\$</span> mutual inductance for stator windings, <span class="math-container">\$\psi_m\$</span> magnetic flux of permanent magnet,<span class="math-container">\$\theta\$</span> rotor position i.e. angle between axes of stator phase <span class="math-container">\$a\$</span> and permanent magnet axes <span class="math-container">\$d\$</span>) <a href="https://i.stack.imgur.com/6m2HF.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6m2HF.jpg" alt="enter image description here"></a></p> <p><strong>2. Clarke transform</strong> (<span class="math-container">\$\hat{a}=e^{j\frac{2\pi}{3}}\$</span>)</p> <p><a href="https://i.stack.imgur.com/M2WxB.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/M2WxB.jpg" alt="enter image description here"></a></p> <p><strong>3. Magnetic fluxes in <span class="math-container">\$(\alpha,\beta)\$</span></strong></p> <p><a href="https://i.stack.imgur.com/oWFma.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/oWFma.jpg" alt="enter image description here"></a></p> <p><strong>4. Park transform</strong></p> <p><a href="https://i.stack.imgur.com/s0tKk.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/s0tKk.jpg" alt="enter image description here"></a></p> <p><strong>5. Magnetic fluxes in <span class="math-container">\$(d,q)\$</span></strong></p> <p><a href="https://i.stack.imgur.com/glhHD.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/glhHD.jpg" alt="enter image description here"></a></p> <p>Based on the formulas for magnetic fluxes in <span class="math-container">\$(d,q)\$</span> I have following formulas <span class="math-container">\$L_d=\frac{2}{3}(L-M)\$</span> and <span class="math-container">\$L_q=\frac{2}{3}(L-M)\$</span>. 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.</p>