A new approach for torque ripple reduction in a faulty surface permanent magnet synchronous motor by inverse current injection

Abstract

The aim of this paper is to present a new method to generate stator currents supplying a surface permanent magnets synchronous motor (SPMSM) presenting an asymmetry of the stator windings. This asymmetry may be due to a lack of turns in one of the stator windings or to an inter-turns short circuit. An analytical model of the SPMSM is developed to calculate the current which allows minimizing the torque ripple generated by the asymmetry. This model is based on the combination of the space harmonics of the flux densities of stator and rotor for the generation of the torque. Our study leads to define a stator inverse current system able to compensate the harmonic torque component at the double of the supply frequency. In the experimental tests, several cases of supply were applied and the results confirmed those obtained by the analytical study essentially concerning the amplitude of the inverse current.

Appendices

Appendix 1

See Tables 2 and 3.

Table 2 SPMSM parameter simulation

Table 3 Rotor flux density space harmonic

Appendix 2

The strategy for calculation of stator currents consists in canceling the space harmonic of stator mmf of rank $h = -p$ in the case of a stator fault in one phase winding. The computation of a mmf space harmonic component is done by considering the currents in each slot. The following developments consider a p pole pair machine with an inter-turn short-circuit fault that occurs in slots ${j}_{1}$ and ${j}_{2}$. K is the ratio of faulty windings in the faulty slots, and ${I}_{{a}1}$ is the rms value of the current in the faulty turns.

Let us consider the mmf ${\varepsilon }_{j}^\mathrm{s}$ generated by the ${n}^{{e}}$ wire placed in slot j. The waveform of ${\varepsilon }_{j}^\mathrm{s}$ is shown in Fig. 19.

Fig. 19

mmf generated by the wires in one slot

${\varepsilon }_{j}^\mathrm{s}$ can be written with a Fourier series as follows:

$$\begin{aligned} {\varepsilon }_{j}^\mathrm{s} =\frac{{n}_{e} {i}_{j}^\mathrm{s} }{(\pi -\delta )}\sum _{{h}=1}^{+\infty } {\varGamma _{h} \frac{{\sin }\left( {{h}({\alpha }^\mathrm{s}-{\beta }_{j}^\mathrm{s} )} \right) }{{h}}} =\sum _{{h}=1}^{+\infty } {{\varepsilon }_{{j},{h}}^\mathrm{s} } \end{aligned}$$

The analytical computation is performed using the complex notation of the mmf components when the SPMSM is assumed to be supplied with sine currents: ${i}_{j}^\mathrm{s} =\sqrt{{2}}{ I}_{j}^\mathrm{s} {\sin }(\omega {t}+{\varphi }_{j})$.

Then, the complex quantity ${\overline{\varepsilon }}_{{j},{h}}^\mathrm{s}$ is defined such that:

$$\begin{aligned} {\varepsilon }_{{j},{h}}^\mathrm{s} = \mathfrak {R}\left[ {{\overline{\varepsilon }}_{{j},{h}}^\mathrm{s} {{e}}^{{i}(\omega {t}-{h}{\alpha }^\mathrm{s})}} \right] \end{aligned}$$

And the rank h of mmf harmonic generated by all the slots can be expressed as:

$$\begin{aligned} {\varepsilon }_{{h}}^\mathrm{s} =\sum _{{j}=1}^{{N}_{{s}}^{{t}}} {{\varepsilon }_{{j},{h}}^\mathrm{s} } =\mathfrak {R}\left[ {\mathrm{e}^{{i}(\omega {t}-{h}{\alpha }^\mathrm{s})}\sum _{{j}=1}^{{N}_{{t}}^\mathrm{s} } {{\overline{\varepsilon }}_{{j},{h}}^\mathrm{s} } } \right] \end{aligned}$$

Case of inter-turn short-circuit fault in stator winding

The stator winding mmf at $h=-p$ is computed in two cases:

• healthy stator winding supplied by direct current of ${I}_\mathrm{d}$ rms value, in this case the mmf harmonic at $h=-p$ is null, because in healthy case only the ranks $h=p(6k+1)$ exist.

  $$\begin{aligned} {\overline{\varepsilon }}_{{d}-{p}}^\mathrm{s} =\sum _{{j}=1}^{{N}_\mathrm{t}^\mathrm{s} } {\frac{{n}^{e} {I}_\mathrm{d}}{\sqrt{{2}}({\pi }-\delta )(-{p})}}{\varGamma }_{-{p}}\mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{dj}} )} =0\nonumber \\ \end{aligned}$$

  (15)

• faulty stator winding supplied by direct current of ${I}_\mathrm{d}$ rms value. In this case, ${Kn}^{{e}}$ wires of the slots ${j}_{1}$ an ${j}_{2}$ are flowed through by a current of ${I}_{{a}1}$ magnitude and the other wires of the both slots ($(1-{K}){n}^{{e}}$ wires) are flowed through by the line current of ${I}_{{d}}$ magnitude:

  $$\begin{aligned} {\overline{\varepsilon }}_{{d},-{p}_{(\mathrm{fault})} }^\mathrm{s}= & {} {\mathop {\mathop {\sum }\limits ^{{N}_\mathrm{t}^\mathrm{s}}_{{j}=1}}\limits _ {{j}\ne {j}_{1} ,{j}_{2}}} \frac{{n}^{e} {I}_\mathrm{d}}{\sqrt{{2}}({\pi }-\delta )(-{p})}{\varGamma _{-{p}} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} + {\varphi }_{{dj}} )}} \nonumber \\&+\, \sum _{{j}={j}_{1} ,{j}_{2} } {\frac{{Kn}^{{e}}{I}_{{a}1} }{\sqrt{{2}}({\pi }-\delta )(-{p})}{\varGamma }_{-{p}} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi } _{{a1dj}} )}} \nonumber \\&+\,\sum _{{j}={j}_{1} ,{j}_{2}} {\frac{(1-{K}){n}^{{e}}{I}_\mathrm{d} }{\sqrt{{2}}({\pi } -\delta )(-{p})}}\varGamma _{-{p}} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi } _{{dj}} )}\nonumber \\ \end{aligned}$$

  (16)

For the stator winding supplied by inverse current of ${I}_{i}$ rms value, it is supposed that the inverse current does not flow through the fault resistance ${r}_{{f}}$, the stator winding mmf at $h=-p$ can be expressed as follows:

$$\begin{aligned} {\overline{\varepsilon }}_{{i},-{p}(\mathrm{fault})}^\mathrm{s} =\sum _{{j}=1}^{{N}_\mathrm{t}^\mathrm{s}} {\frac{{n}^{e} {\overline{{I}}}_{i} }{\sqrt{{2}}({\pi }-{\delta })(-{p})}\varGamma _{-{p}} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} + \varphi _{{ij}} )}} \end{aligned}$$

(17)

To cancel the harmonic at $h = -p$, the mmf generated by the inverse and direct current must satisfy the following equation:

$$\begin{aligned} {\overline{\varepsilon }}_{{i},-{p}(\mathrm{fault})}^\mathrm{s} ={\overline{\varepsilon }}_{{d},-{p}}^\mathrm{s} -{\overline{\varepsilon }}_{{d},-{p}(\mathrm{fault})}^\mathrm{s} \end{aligned}$$

(18)

Then it comes:

$$\begin{aligned} {\overline{\varepsilon }}_{{i},-{p}_{(\mathrm{fault})}}^\mathrm{s}= & {} \sum _{{j}={j}_{1} ,{j}_{2} } \frac{{n}^{{e}}}{\sqrt{{2}}(\pi -\delta )(-{p})}\varGamma _{-{p}} \left( {I}_\mathrm{d} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi } _{{dj}} )}\right. \\&\left. -\,{ KI }_{{a}1} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{a1dj}} )}-(1-{K}){I}_\mathrm{d} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{dj}} )} \right) \end{aligned}$$

Replacing ${\varepsilon }_{{i},-{p}(\mathrm{fault})}^\mathrm{s}$ with (17), it comes:

$$\begin{aligned}&\sum _{{j}=1}^{{N}_\mathrm{t}^\mathrm{s} } {\frac{{n}^{{e}}{\overline{{I}}_{i}} }{\sqrt{{2}}({\pi }-\delta )(-{p})}{\varGamma }_{-{p}} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{ij}} )}}\nonumber \\&\quad =\sum _{{j}={j}_1 ,{j}_{2} } \frac{{n}^{e} }{\sqrt{{2}}(\pi -\delta )(-{p})}\varGamma _{-2} \left( {I}_\mathrm{d} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{dj}} )}\right. \nonumber \\&\quad \quad \left. -\,{ KI }_{{a}1} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{a1dj}} )}-(1-{K}){I}_\mathrm{d} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi } _{{dj}} )} \right) \end{aligned}$$

(19)

Finally, after simplification, the relationship between the direct current and the inverse current used to compensate the harmonic of rank $h=-p$ is given by:

$$\begin{aligned} {\overline{{I}}}_{i} =\frac{\sum \nolimits _{{j}={j}_1 ,{j}_{2} } \left( {{I}_\mathrm{d} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{dj}} )}} -{K}{I}_{{a}1} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi } _{{a1dj}})}-(1-{K}){I}_\mathrm{d} \mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{dj}} )}\right) }{\sum \nolimits _{{j}=1}^{{N}_\mathrm{t}^\mathrm{s} } {\mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{ij}} )}} }\nonumber \\ \end{aligned}$$

(20)

And finally:

$$\begin{aligned} {\overline{{I}}}_{i} ={K}\frac{{I}_\mathrm{d} \sum \nolimits _{{j}={j}_{1} ,{j}_{2} } {\mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{dj}} )}} -{I}_{{a}1} \sum \nolimits _{{j}={j}_1 ,{{j}}_{2} } {\mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{a1dj}} )}} }{\sum \nolimits _{{j}=1}^{{N}_\mathrm{t}^\mathrm{s} } {\mathrm{e}^{{i}(-{p}{\beta }_{j}^\mathrm{s} +{\varphi }_{{ij}} )}} }\nonumber \\ \end{aligned}$$

(21)