Miniature magneto-oscillatory wireless sensor for magnetic field and gradient measurements

Abstract

Magneto-oscillatory devices have been recently developed as very potent wireless miniature position trackers and sensors with an exceptional accuracy and sensing distance for surgical and robotic applications. However, it is still unclear to which extend a mechanically resonating sub-millimeter magnet interacts with external magnetic fields or gradients, which induce frequency shifts of sub-mHz to several Hz and therefore affect the sensing accuracy. Here, we investigate this effect experimentally on a cantilever-based magneto-oscillatory wireless sensor (MOWS) and build an analytical model concerning magnetic and mechanical interactions. The millimeter-scale MOWS is capable to detect magnetic fields with sub-uT resolution to at least +/- 5 mT, and simultaneously detects magnetic field gradients with a resolution of 65 uT/m to at least +/- 50 mT/m. The magnetic field sensitivity allows direct calculation of mechanical device properties, and by rotation, individual contributions of the magnetic field and gradient can be analyzed. The derived model is general and can be applied to other magneto-oscillatory systems interacting with magnetic environments.

Figures (4)

• Figure 1: (a) Schematic overview of the system to scale with an enlarged side view of the MOWS in its intrinsic coordinate system. The frequency of the magnets oscillation depends on the magnetic field $B$ and gradient $G$ in the main direction $x'$. (b) Physical sub-models of the cantilever oscillator with the deflection angle $\uptheta_t$. (c) Time sequence of a signal acquisition with an excitation coil current $I_\mathrm{coil}$ to excite the magnet's oscillation. Two magnetometers $S_1$ and $S_2$ pick up the MOWS signal which is used for evaluation, shown as a shaded area. (d) Exemplary measured MOWS signal recorded at an effective distance of 6.75 cm and its corresponding signal fit.
• Figure 2: (a) Schematic experimental setup from the top view in global coordinates and two coils at 45° angles for excitation of the MOWS. (b) Examples of magnetic signals $S_\mathrm{final}$ with a maximum deflection angle of 15° and a resonance frequency of 100 Hz, normalized by parameter $\kappa$, see Eq. (S12), stemming from the MOWS at three different offset angles $\uptheta_0$.
• Figure 3: (a) Magnetic field $B$ versus resonance frequency shift $\Delta f_{B,0}$ for sub-$\upmu$T magnetic fields applied in intrinsic $x'$-direction. The red line shows the linear correlation between both parameters. (b) $B$ versus $\Delta f_{B,0}$ for $\upmu$T-range magnetic fields. (c) $B$ versus resonance frequency $f_{B,0}$ for mT-range magnetic fields with the physical model fit according to Eq. (\ref{['eq:delta_f']}) without gradient contributions. Black lines indicate limit values of the model. (d) $B$ versus the theoretical field sensitivity $\upbeta$ according to Eq. (\ref{['eq:beta']}). (e) Enlarged plot of $B$ versus $\Delta f_{B,0}$ for mT-range magnetic fields.
• Figure 4: Polar plots for mapping of parameters at various conditions by rotations about the $z$-axis by angle $\uptheta_0$. Only absolute values are plotted for readability and the actual corresponding signs are labeled in the plot. Linear versions of the plots are shown in Fig. S5. (a) Frequency shift $\Delta f_{B,0}$ in a homogeneous magnetic field applied in $x$-direction (purple arrow) without gradient, with a sine fit curve and a separate ground truth measurement (red) using a wired magnetometer at the same location. The green arrow indicates the environmental magnetic field. (b) Frequency shift $\Delta f_{B,0}$ in the environmental magnetic field, with sine fit curve and ground truth measurement. (c) Frequency shift $\Delta f_{B,G}$ in a near-zero magnetic field with a strong gradient applied in $-x$-direction, with fit curve Eq. (\ref{['eq:decompose_fit']}), and the corresponding decomposition for the magnetic field $B$ and gradient $G$. (d) Frequency shift $\Delta f_{B,G}$, induced by a magnetic source, with fit curve and the corresponding decomposition.