Rotor instability due to electromechanical interactions in synchronous machines

Abstract

For winding configurations with parallel paths, synchronous electric motors can exhibit large amplitude rotor oscillations, which cannot be related to any rotation order nor investigated with classical harmonic analysis. This unusual behaviour is critical, since it can cause the rotor to hit the stator. It is a direct consequence of the coupling of rotor displacement with the magnetic flux and current distribution. It depends not only on the winding configuration, but also on the various motor design parameters. In order to explain this phenomenon, a simple analytical motor model has been built, including the full coupling between mechanical and electrical equations, which reproduces the behaviour observed experimentally. Using non-dimensional expressions allows to extract the few design ratios characterising the system, whose influence is then explored by stability analysis, in order to set their stability domain. It becomes thus possible to determine which parameters' configuration would lead to an unstable behaviour.

Appendix

The magnetomotive ${mmf}_{stator}$ force is uniquely defined along the stator bore as a periodic step function by the equations

$$\left\{\begin{array}{c}{\mathbf{m}\mathbf{m}\mathbf{f}}_{{s}{t}{a}{t}{o}{r}\;{{{i}}+1}}-{\mathbf{m}\mathbf{m}\mathbf{f}}_{{s}{t}{a}{t}{o}{r}\;{{i}}}={{N}}\cdot {\mathbf{I}}_{{s}{{i}}}\;{{f}}{{o}}{{r}}\;{{a}}{{n}}{{y}}{\; {i}}=1,2..{{{n}}}_{{{s}}}\\\sum_{{{i}}=1}^{{{{n}}}_{{{s}}}}{\mathbf{mmf}}_{{stator}\;{{i}}}=0\end{array}\right.$$

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each step corresponding to a slot i travelled by a total current $N\cdot {\mathbf{I}}_{\mathbf{s} i}$ (Fig. 6). One verifies that the expression of the stator mmf given in Eq. (9) and Eq. (10) fulfils both equations:

$${\mathbf{m}\mathbf{m}\mathbf{f}}_{{stator}}=N\cdot {\mathbf{N}}_{\mathbf{m}\mathbf{m}\mathbf{f}}\cdot {\mathbf{I}}_{\mathbf{s}},$$

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where the n_s x n_s matrix N_mmf is:

$$ {\mathbf{N}}_{{{\mathbf{mmf}} r,q}} = \frac{1}{{n_{s} }} \cdot \left\{ {\begin{array}{*{20}c} {q, {\text{if}}\; q < r} \\ {q - n_{s} , {\text{if}}\; q \ge r} \\ \end{array} } \right.\; {\text{with}} \; r \; {\text{and }}\;q = 1,2 \ldots n_{s}. $$

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$$\begin{aligned}&{\mathbf{m}\mathbf{m}\mathbf{f}}_{{stator}\;{ i+1}}-{\mathbf{m}\mathbf{m}\mathbf{f}}_{{stator} \;i}\\&\quad=N\cdot \sum_{q=1}^{{n}_{s}}{\mathbf{N}}_{\mathbf{m}\mathbf{m}\mathbf{f} i+1, q}\cdot {\mathbf{I}}_{\mathbf{s} q}-N\cdot \sum_{q=1}^{{n}_{s}}{\mathbf{N}}_{\mathbf{m}\mathbf{m}\mathbf{f} i, q}\cdot {\mathbf{I}}_{\mathbf{s} q}\\&\quad=N\cdot \sum_{q=1}^{{n}_{s}}\left({\mathbf{N}}_{\mathbf{m}\mathbf{m}\mathbf{f} i+1, q}-{\mathbf{N}}_{\mathbf{m}\mathbf{m}\mathbf{f} i, q}\right)\cdot {\mathbf{I}}_{\mathbf{s} q}\\&\quad=N\cdot \sum_{q=i}\left({\mathbf{N}}_{\mathbf{m}\mathbf{m}\mathbf{f} i+1, q}-{\mathbf{N}}_{\mathbf{m}\mathbf{m}\mathbf{f} i, q}\right)\cdot {\mathbf{I}}_{\mathbf{s} q}\\&\quad=N\cdot \left({\mathbf{N}}_{\mathbf{m}\mathbf{m}\mathbf{f} i+1, i}-{\mathbf{N}}_{\mathbf{m}\mathbf{m}\mathbf{f} i, i}\right)\cdot {\mathbf{I}}_{\mathbf{s} i}\\&\quad=\frac{N}{{n}_{s}}\cdot \left(i-\left(i-{n}_{s}\right)\right)\cdot {\mathbf{I}}_{\mathbf{s} i}\\&\quad=N\cdot {\mathbf{I}}_{\mathbf{s} i}.\end{aligned}$$

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$$\begin{aligned}\begin{aligned}\sum_{{{i}}=1}^{{{{n}}}_{{{s}}}}{\mathbf{m}\mathbf{m}\mathbf{f}}_{{s}{t}{a}{t}{o}{r}\;{{i}}}&=\sum_{{{i}}=1}^{{{{n}}}_{{{s}}}}{f{N}}\cdot \sum_{{{q}}}{\mathbf{N}}_{\mathbf{mmf}\; i,q}\cdot {\mathbf{I}}_{\mathbf{s}{{q}}}\\ &={{N}}\cdot \sum_{{{q}}=1}^{{{{n}}}_{{{s}}}}{\mathbf{I}}_{\mathbf{s}{{q}}}\cdot \sum_{{{i}}}{\mathbf{N}}_{\mathbf{mmf}\; i,q}\end{aligned}\end{aligned}$$

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from ${\mathbf{N}}_{\mathbf{m}\mathbf{m}\mathbf{f}}$ definition above it follows:

$$\begin{aligned}\sum\limits_{i} {{\mathbf{N}}_{{{\mathbf{mmf}}\;i,q}} } &= \sum\limits_{{i = 1}}^{q} {{\mathbf{N}}_{{{\mathbf{mmf}}\;i,q}} } + \sum\limits_{{i = q + 1}}^{{n_{s} }} {{\mathbf{N}}_{{{\mathbf{mmf}}\;i,q}} } \\& = q \cdot \left( {q - n_{s} } \right) + \left( {n_{s} - q} \right) \cdot q = 0\end{aligned} $$

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