Abstract
Inductive sensors play a pivotal role in online oil debris detection and fault diagnosis. Inductive rotary coils are generally used in debris detection but lack high particle detection sensitivity. This paper proposes a shape optimization method to improve detection sensitivity and enhance sensor performance. The relative coil density is introduced as a shape parameter to quantify the geometry of the rotary coil and is decoupled from the coil size and wire arrangement. The signal formulas of arbitrarily shaped rotary coils are derived based on the relative coil density and reveal the effect of several coil parameters. This study provides general optimization methods for enhancing performance for application to microparticle inductance and resistance detection. To illustrate the benefit of the coil optimization method, the optimal coil shapes of various application cases are numerically designed and discussed. This optimization framework can be easily generalized to other application scenarios of rotary coils requiring optimization of magnetic field distribution.
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Abbreviations
- \({\varvec{r}}\), \(\widetilde{{\varvec{r}}}\) :
-
The position vectors of arbitrary point in coil space and characteristic detection point
- \({R}_{p}({\varvec{r}})\), \({L}_{p}({\varvec{r}})\) :
-
The resistance and inductance changes when the passing debris is at position \({\varvec{r}}\)
- \({\varvec{h}}\left({\varvec{r}}\right)\), \({\varvec{a}}({\varvec{r}})\) :
-
The magnetic strength and magnetic vector potential applied by a coil loaded with the direct unit current at \({\varvec{r}}\)
- \({\mu }_{0}\) :
-
Vacuum permeability \(\mu_{0} \, = \,4\pi \, \times \,10^{ - 7} \,H/m\)
- \({\eta }_{R}, {\eta }_{L}\) :
-
The coil basic inductance and resistance, \( \begin{array}{*{20}c} {\eta _{R} = R_{p} /R_{c} \eta _{L} = L_{p} /L_{c} } \\ \end{array} \)
- \({{\varvec{D}}}_{1}\) :
-
3D-local cylindrical coordinate space
- \({{\varvec{D}}}_{2}\) :
-
2D-local Cartesian coordinate space
- \(\widehat{{\varvec{\theta}}}\) :
-
The unit angular direct vector round z-axis
- \({{\varvec{\Omega}}}_{{\varvec{E}}}\) , \({{\varvec{\Omega}}}_{{\varvec{C}}}\) , \({\boldsymbol{ }\boldsymbol{ }\boldsymbol{\Omega }}_{{\varvec{F}}},\boldsymbol{ }\boldsymbol{ }{{\varvec{\Omega}}}_{{\varvec{O}}}\) :
-
The empty region, the region filled with coil wires, the region of flow path, and the design region
- \({\varvec{C}}\left({\varvec{r}}\right)\), \(C({\varvec{r}})\) :
-
The coil vector density at \({\varvec{r}}\) and the coil scalar density at \({\varvec{r}}\)
- \(C_{\max }\) :
-
The maximum coil density when coil wires are tightly wound
- \(c\left({\varvec{r}}\right)\) :
-
The relative coil density \(c({\varvec{r}})\)
- \({R}_{c}\left(c\right)\),\({ L}_{c}\left(c\right)\) :
-
The resistance and inductance of empty coil
- \(\omega \) :
-
Angular frequency of the excitation field
- \(\epsilon ,\beta \) :
-
Normalized \({R}_{2}\) and \(H\), namely, \(\frac{{R}_{2}}{P}\) and \(\frac{L}{P}\)
- \(\gamma ,\xi \) :
-
Normalized \(r\) and \(z\), namely, \(\frac{r}{P}\) and \(\frac{z}{P}\)
- \({v}_{p}\) :
-
Particle volume
- \({L}_{c}, {R}_{c}\) :
-
Empty-coil inductance and resistance
- \(\chi \) :
-
Magnetic susceptibility in an alternating magnetic field
- \(L\) :
-
Half of the winding length along the coil axis
- \({l}_{w}\) :
-
Total length of coil wire
- \({\sigma }_{m}\) :
-
Conductivity of metal
- \({\delta }_{m}\) :
-
Skin-depth of metal wire
- \({R}_{1}{, R}_{2}\) :
-
Inner radius and outer radius of the coil
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Acknowledgements
The authors are grateful for the support from Shandong Provincial Key Research and Development Plan (No. 2021CXGC010702), National Natural Science Foundation of China (No. 51909047).
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Xue, B., Yu, C., Hua, G. et al. Arbitrary-cross-section rotary inductive coils for microparticle detection: analytical modeling, optimization and design. Struct Multidisc Optim 66, 154 (2023). https://doi.org/10.1007/s00158-023-03606-9
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DOI: https://doi.org/10.1007/s00158-023-03606-9