High-precision wireless displacement monitoring microsystem based on TMR effect for structural health monitoring

Abstract

This study presents a high-precision wireless displacement monitoring microsystem that utilizes the tunnel magnetoresistance (TMR) effect for structural health monitoring (SHM). The system overcomes limitations of traditional SHM methods, providing high-precision, intelligent and lightweight measurements. We established an analytical model of magnetic field displacement and optimized its linear range. Considering the measurement error caused by magnetic field decay, we designed an adaptive sensitivity correction method, thus avoiding the tedious magnetic field numerical fitting process. The system’s accuracy and stability are validated through comparison with laser ranging, showing high accuracy within the range of ±7.5 mm, a resolution of 0.4 μm, and a long-term working accuracy better than 2.25 μm. The core system is less than 3.84 cm³ in size and is inexpensive to manufacture, making it ideal for mass deployment across a broad range of infrastructure. This work outperforms other state-of-the-art methods in the field in terms of accuracy, cost, size, and power consumption. Practical applications in monitoring concrete deformation, crack width changes, and bridge beam end slip deformation highlight its versatility and effectiveness. Its stable performance in long-term autonomous operation has also been verified in a 39-day actual bridge test, making it a valuable tool for enhancing infrastructure maintenance and safety.

Introduction

The integrity of civil infrastructure, including buildings and bridges, is paramount to public safety and economic stability^1,2. These structures face numerous threats, ranging from environmental factors to dynamic loads, which can lead to distress such as cracks, settlements, and slips^3–5. Therefore, real-time monitoring and evaluation of key parameters of these structures is essential to assess their health and prevent accidents. Structural health monitoring (SHM) systems have emerged as a vital tool in this regard, enabling the detection of early signs of degradation and the quantification of structural responses to various stimuli^6–8.

Traditional SHM techniques, while valuable, often rely on labor-intensive methods or are constrained by limitations in sensitivity, spatial resolution, or the ability to provide continuous monitoring. For example, optical methods have the highest accuracy when measuring displacement, but are often hampered by environmental conditions due to their reliance on expensive instrumentation and auxiliary facilities^9,10. GNSS has the potential to be deployed at multiple points, but its accuracy is the worst¹¹. Some smart devices need to be embedded in the structure¹², which is very troublesome for existing bridges or buildings. Vibration data are common in SHM, but they do not directly reflect displacement^13,14. There are also some visual measurement solutions, but they are not completely satisfactory in terms of measurement accuracy and continuous monitoring^15,16.

The advent of advanced sensor technologies, especially micro-electromechanical system (MEMS) technologies, such as flexible sensitive materials^17,18, capacitive sensors^19,20, and magnetometers, has marked a significant advance in the field of SHM. Among them, magnetically sensitive materials that are compatible with MEMS processes, including Hall components, anisotropic magnetoresistance (AMR), giant magnetoresistance (GMR) and tunnel magnetoresistance (TMR) materials²¹, offer low-cost, miniaturized, non-intrusive displacement, deformation and attitude measurement capabilities and have the potential to be deployed on a large scale without causing significant disruption to the operation of the structure. For example, over the past two decades, these sensors have been widely used in non-destructive SHM^22–24, biomedical^25,26, robotics²⁷, and aerospace²⁸.

In recent years, with the maturity of TMR sensor technology, due to its advantages of high sensitivity, high precision and low power consumption, it is more suitable for long-term, autonomous, precise and quantitative monitoring of infrastructure structures²⁹. Although some of the most advanced Hall-effect sensors are comparable to TMR sensors in accuracy and sensitivity³⁰, their power consumption is still a significant disadvantage. However, in previous work, high-precision displacement measurement based on TMR often relies on prefabricated high-precision magnetic scales³¹ or complex numerical fitting models to map magnetic field to displacement^32,33, and do not take into account possible magnetic field decay, which will bring difficulties to practical applications.

Here, we propose a wireless high-precision displacement sensing microsystem (Fig. 1). based on TMR sensors. When the position of the TMR sensor in the magnetic field changes, the displacement change is calculated based on the measured magnetic field change (Fig. 1a, b). Compared with previous displacement measurement methods^31–34, we have established a complete magnetic field analytical model and optimized the parameters of the discovered linear working range to improve accuracy. Considering the measurement error caused by magnetic field decay, we designed an adaptive sensitivity correction method, thus avoiding the tedious magnetic field numerical fitting process. At the same time, we integrated the TMR sensor into the microsystem for the first time (Fig. 1c, d). The entire microsystem includes control modules, sensor modules, energy modules and wireless modules to meet the needs of self-powered intelligent monitoring. Different modules will be processed into independent layers of high-density circuits, which are interconnected in a vertical stacking manner to achieve high-density size and customizable function integration. Finally, the size of the entire system is 20 mm × 16 mm × 12 mm, and the cost of core components does not exceed US$20, which has valuable prospects for widespread deployment of large infrastructure.

Fig. 1. Wireless displacement monitoring microsystem.

a The relative position of the TMR sensor and the magnetic field changes. b The magnetic field change and displacement change show a linear approximate relationship. c Framework schematic diagram of the microsystem, where different functional modules are integrated in a vertical stacking manner. d Physical schematic diagram of the vertically stacked integration of the microsystem

Theoretical model and optimization

Sensing principle

To determine the magnetic field applied to the TMR sensor, the spatial distribution of the magnetic field generated by the permanent magnet is studied. Several methods, including the molecular circulation model and the finite element method, have been proposed. Based on the molecular circulation assumption describing the magnetic medium, the spatial magnetic field is generated by the surface current of the permanent magnet³⁵. Therefore, the analytical expression for the spatial magnetic field distribution outside the permanent magnet can be derived using the Biot-Savart theorem:

$$\mathit{B}\left(x,y,z\right)=\frac{{\mu }_0}{4\pi }{\int }_0^h\oint \frac{J_{\mathrm{m}}\mathrm{d}\mathit{l}\times \left(\mathit{r}-{\mathit{r}}_{\mathrm{m}}\right)}{{\left|\mathit{r}-{\mathit{r}}_{\mathrm{m}}\right|}^3}\mathrm{d}{z}_{\mathrm{m}},$$ 1

where $\mathit{r}$ and ${\mathit{r}}_{\mathrm{m}}$ are the vectors of the TMR sensing position and the permanent magnet surface integrated position respectively, $J_{\mathrm{m}}=B_{\mathrm{r}}/{\mu }_0$ is the assumed permanent magnet surface current density, $B_{\mathrm{r}}$ is the residual magnetic induction, and ${\mu }_0$ is the vacuum permeability.

For a rectangular permanent magnet with a length, width and height of $a$, $b$, and $h$ respectively as shown in Fig. 2a, the molecular current loop integral at $z=z_{\mathrm{m}}$ can be expressed as

$$\begin{array}{c}\hfill \oint \frac{\mathrm{d}\mathit{l}\times \left(\mathit{r}-{\mathit{r}}_{\mathrm{m}}\right)}{{∣\mathit{r}-{\mathit{r}}_{\mathrm{m}}∣}^3}={\int }_0^a{\left(\frac{\mathrm{d}{\mathit{x}}_{\mathrm{m}}\times \left(\mathit{r}-{\mathit{r}}_{\mathrm{m}}\right)}{{∣\mathit{r}-{\mathit{r}}_{\mathrm{m}}∣}^3}\right.∣}_{y_{\mathrm{m}}=0}+{\int }_0^b{\left(\frac{\mathrm{d}{\mathit{y}}_{\mathrm{m}}\times \left(\mathit{r}-{\mathit{r}}_{\mathrm{m}}\right)}{{∣\mathit{r}-{\mathit{r}}_{\mathrm{m}}∣}^3}\right.∣}_{x_{\mathrm{m}}=a}\hfill \\ \hfill +{\int }_a^0{\left(\frac{-\mathrm{d}{\mathit{x}}_{\mathrm{m}}\times \left(\mathit{r}-{\mathit{r}}_{\mathrm{m}}\right)}{{∣\mathit{r}-{\mathit{r}}_{\mathrm{m}}∣}^3}\right.∣}_{y_{\mathrm{m}}=b}+{\int }_b^0{\left(\frac{\mathrm{d}{\mathit{y}}_{\mathrm{m}}\times \left(\mathit{r}-{\mathit{r}}_{\mathrm{m}}\right)}{{∣\mathit{r}-{\mathit{r}}_{\mathrm{m}}∣}^3}\right.∣}_{x_{\mathrm{m}}=0}\hfill \\ \hfill ={\int }_0^a\frac{\left(0,z-z_{\mathrm{m}},-y\right)d{x}_{\mathrm{m}}}{{\left[{\left(x-x_{\mathrm{m}}\right)}^2+y^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}+{\int }_0^b\frac{\left(z_{\mathrm{m}}-z,0,x-a\right)d{y}_{\mathrm{m}}}{{\left[{\left(x-a\right)}^2+{\left(y-y_{\mathrm{m}}\right)}^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}\hfill \\ \hfill +{\int }_0^a\frac{\left(0,z-z_{\mathrm{m}},b-y\right)d{x}_{\mathrm{m}}}{{\left[{\left(x-x_{\mathrm{m}}\right)}^2+{\left(y-b\right)}^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}+{\int }_0^b\frac{\left(z_{\mathrm{m}}-z,0,x\right)d{y}_{\mathrm{m}}}{{\left[x^2+{\left(y-y_{\mathrm{m}}\right)}^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}\hfill \end{array}$$ 2

Fig. 2. Mechanism of the measurement method.

a Analytical model of magnetic field of rectangular permanent magnet based on the molecular current loop model. b Magnetic field results obtained from FES, molecular current model and its linear approximation

Based on eq. (2), the direction components of the magnetic field at the TMR sensing position can be given as:

$$\begin{array}{c}\left\{\begin{array}{c}{B}_x\left(x,y,z\right)=\frac{J_{\mathrm{m}}}{4\pi }{\int }_0^h\left\{{\int }_0^b\frac{\left(z_{\mathrm{m}}-z\right)d{y}_{\mathrm{m}}}{{\left[{\left(x-a\right)}^2+{\left(y-y_{\mathrm{m}}\right)}^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}+{\int }_0^b\frac{\left(z_{\mathrm{m}}-z\right)d{y}_{\mathrm{m}}}{{\left[x^2+{\left(y-y_{\mathrm{m}}\right)}^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}\right\}d{z}_{\mathrm{m}}\\ B_y\left(x,y,z\right)=\frac{J_{\mathrm{m}}}{4\pi }{\int }_0^h\left\{{\int }_0^a\frac{\left(z-z_{\mathrm{m}}\right)d{x}_{\mathrm{m}}}{{\left[{\left(x-x_{\mathrm{m}}\right)}^2+y^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}+{\int }_0^a\frac{\left(z-z_{\mathrm{m}}\right)d{x}_{\mathrm{m}}}{{\left[{\left(x-x_{\mathrm{m}}\right)}^2+{\left(y-b\right)}^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}\right\}d{z}_{\mathrm{m}}\\ B_z\left(x,y,z\right)=\frac{J_{\mathrm{m}}}{4\pi }{\int }_0^h\left\{\begin{array}{c}{\int }_0^a\frac{-\mathrm{y}d{x}_{\mathrm{m}}}{{\left[{\left(x-x_{\mathrm{m}}\right)}^2+y^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}+{\int }_0^a\frac{\left(b-y\right)d{x}_{\mathrm{m}}}{{\left[{\left(x-x_{\mathrm{m}}\right)}^2+{\left(y-b\right)}^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}\\ +{\int }_0^b\frac{\left(x-a\right)d{y}_{\mathrm{m}}}{{\left[{\left(x-a\right)}^2+{\left(y-y_{\mathrm{m}}\right)}^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}+{\int }_0^b\frac{\mathit{xd}{y}_{\mathrm{m}}}{{\left[x^2+{\left(y-y_{\mathrm{m}}\right)}^2+{\left(z-z_{\mathrm{m}}\right)}^2\right]}^{\frac{3}{2}}}\end{array}\right\}\end{array}.\right)\mathrm{\#}\end{array}$$ 3

Two operators are introduced as follows:

$$\mathrm{\varphi }\left(p_1,p_2,p_3\right)=\arctan \frac{p_1\left(p_3-z_{\mathrm{m}}\right)}{p_2\sqrt{p_1^2+p_2^2+{\left(p_3-z_{\mathrm{m}}\right)}^2}},$$ 4.1

$$\mathrm{\Gamma }\left(q_1,q_2,q_3\right)=\ln \frac{\sqrt{q_1^2+q_2^2+{\left(q_3-z_{\mathrm{m}}\right)}^2}-q_2}{\sqrt{q_1^2+q_2^2+{\left(q_3-z_{\mathrm{m}}\right)}^2}+q_2},$$ 4.2

consequently, where

$$\left\{\begin{array}{c}{B}_x\left(x,y,z\right)=-\frac{K}{2}{\left(\left[\mathrm{\Gamma }\left(a-x,y,z\right)+\mathrm{\Gamma }\left(a-x,b-y,z\right)-\mathrm{\Gamma }\left(x,y,z\right)-\mathrm{\Gamma }\left(x,b-y,z\right)\right]\right|}_{z_{\mathrm{m}}=0}^{z_{\mathrm{m}}=h}\hfill \\ B_y\left(x,y,z\right)=-\frac{K}{2}{\left(\left[\mathrm{\Gamma }\left(b-y,x,z\right)+\mathrm{\Gamma }\left(b-y,a-x,z\right)-\mathrm{\Gamma }\left(y,x,z\right)-\mathrm{\Gamma }\left(y,a-x,z\right)\right]\right|}_{z_{\mathrm{m}}=0}^{z_{\mathrm{m}}=h}\hfill \\ B_z\left(x,y,z\right)=-K{\left(\left[\begin{array}{l}\mathrm{\varphi }\left(y,a-x,z\right)+\mathrm{\varphi }\left(b-y,a-x,z\right)+\mathrm{\varphi }\left(x,b-y,z\right)+\mathrm{\varphi }\left(a-x,b-y,z\right)\hfill \\ +\mathrm{\varphi }\left(b-y,x,z\right)+\mathrm{\varphi }\left(x,y,z\right)+\mathrm{\varphi }\left(a-x,y,z\right)+\mathrm{\varphi }\left(y,x,z\right)\hfill \end{array}\right]\right|}_{z_{\mathrm{m}}=0}^{z_{\mathrm{m}}=h}\hfill \end{array},\right)$$ 5

$$K=\frac{{\mu }_0{J}_m}{4\pi }=\frac{B_r}{4\pi }.$$ 6

It can be observed that the expressions of the magnetic field components in the x and y directions are highly similar and are symmetrical about the central axis ($x=\frac{a}{2},y=\frac{b}{2}$) of the permanent magnet. In particular, by letting $y=\frac{b}{2}$ and replacing $x$ with $x+\frac{a}{2}$ to move the origin of the coordinate system to the position of the central axis of the permanent magnet, the magnetic field expression in the x direction is obtained as

$$B_x=K{\left(\left[\ln \frac{\left[\sqrt{{\left(x+\frac{a}{2}\right)}^2+\frac{b^2}{4}+{\left(z-z_{\mathrm{m}}\right)}^2}-\frac{b}{2}\right]\left[\sqrt{{\left(x-\frac{a}{2}\right)}^2+\frac{b^2}{4}+{\left(z-z_{\mathrm{m}}\right)}^2}+\frac{b}{2}\right]}{\left[\sqrt{{\left(x+\frac{a}{2}\right)}^2+\frac{b^2}{4}+{\left(z-z_{\mathrm{m}}\right)}^2}+\frac{b}{2}\right]\left[\sqrt{{\left(x-\frac{a}{2}\right)}^2+\frac{b^2}{4}+{\left(z-z_{\mathrm{m}}\right)}^2}-\frac{b}{2}\right]}\right]\right|}_{z_{\mathrm{m}}=0}^{z_{\mathrm{m}}=h}.$$ 7

The Taylor expansion of the eq. (6) at $x=0$ finally gives the analytical formula for the permanent magnet magnetic field of one end face near the central axis:

$$B_x=K\mathit{abx}\left\{\frac{1}{\left[\frac{a^2}{4}+{\left(z-h\right)}^2\right]\sqrt{\frac{a^2}{4}+\frac{b^2}{4}+{\left(z-h\right)}^2}}-\frac{1}{\left[\frac{a^2}{4}+z^2\right]\sqrt{\frac{a^2}{4}+\frac{b^2}{4}+z^2}}\right\}.$$ 8

It is evident that the magnetic field strength in a certain direction near the central axis of the permanent magnet is linearly related to the displacement in that direction. Using this characteristic, we can calculate the displacement of the TMR sensor relative to the permanent magnet based on the magnetic field strength measured by the sensor. According to equations (6) and (8), the sensitivity of the displacement sensing after linear approximation is given by:

$$K_{\mathrm{s}}=\frac{B_r}{4\pi }\mathit{ab}\left\{\frac{1}{\left[\frac{a^2}{4}+g^2\right]\sqrt{\frac{a^2}{4}+\frac{b^2}{4}+g^2}}-\frac{1}{\left[\frac{a^2}{4}+{\left(h+g\right)}^2\right]\sqrt{\frac{a^2}{4}+\frac{b^2}{4}+{\left(h+g\right)}^2}}\right\}$$ 9

This sensitivity is crucial for determining the displacement of the TMR sensor relative to the permanent magnet, allowing for precise monitoring of structural health with high accuracy.

Analysis and optimization

In the finite element simulation (FES) software COMSOL, we established a numerical model of TMR and permanent magnet magnetic field. At the same time, the analytical result obtained based on the molecular current loop model is similar to the FES results (Fig. 2b), which proves the correctness of the analytical magnetic field model we established. It can be found that when the TMR sensor moves along the x-axis, there is a region where the magnetic field intensity changes approximately linearly, and the magnetic field curve near the zero point can be approximated by a straight line. In this way, the change in magnetic field intensity can be used to directly linearly map the displacement change. In order to measure and reduce the error caused by linearization, we will further optimize the parameters of the sensor system based on the analytical results of the magnetic field, including the length ($a)$, width ($b$) and height ($h$) of the permanent magnet, and the gap (g) from the TMR sensor to the end face of the permanent magnet.

The impact of the gap must be primarily considered. This can be discussed in two parts. First, the distance between the two peak points is related to $a$. As the gap g increases, the peak values will gradually shift outwards from ±a/2 (Figs. 3a and S1a–c in supplementary materials). Upon examining the curves of the analytical results and the linear approximation results, it is observed that when g is very small, the analytical curve will bulge upwards, whereas as g increases, the analytical curve will descend below the straight line and approach it (Fig. S1d in supplementary materials). We provisionally defined the linear working range as [$-\mathrm{a}/2,\mathrm{a}/2$], and upon calculating the error introduced by the linear approximation, it is found that this error is non-monotonic with different $x$ (Fig. S1e in supplementary materials). Simply speaking, the average error of $B_x$ in the working range is as follows: the larger the g, the smaller the average error. However, we also observed that the larger the g, the smaller the sensitivity of $B_x$ to $x$, which is not conducive to high-precision displacement measurement (Fig. S1f in supplementary materials). Therefore, we believe that there is a suitable g that achieves a balance between error and sensitivity.

Fig. 3. Analysis and optimization of system structure parameters.

a $B_x$ curves at different g, where the circles indicate the positions of the peaks and the linear working range is between the two peaks. b Schematic diagram of the linear working range obtained using MEC and AEC. c Changes in ALR according to MEC and AEC when changing g. d 2D intensity diagram of ALR at different a and g. e 2D intensity diagram of ALR at different b and g. f 2D intensity diagram of ALR at different h and g

Furthermore, the maximum error criterion (MEC) and the average error criterion (AEC) are used to determine the allowable linear region (ALR). The difference between the magnetic field analysis result and the linearization result is calculated respectively when the working range extends from 0 to [$-\mathrm{a}/2,\mathrm{a}/2$] or even larger. If the error exceeds the value specified by MEC or AEC, the displacement $x$ here is considered to be ALR. In this work, we controlled the maximum error to be no more than 1% and the average error to be less than 0.5%. With this as the boundary condition, we can determine the maximum linear working area within the allowable error range (Fig. 3b). It can be seen that there is a characteristic g value that makes the ALR reach the local maximum value under the control of MEC and AEC (Fig. 3c). Based on this, the actual g value can be fine-tuned according to application requirements. For example, in scenarios requiring high-sensitivity measurements, the g value can be appropriately reduced; on the other hand, the performance of some magnetometers will be reduced in strong magnetic field environments, so the g value can be appropriately reduced to cope with such situations.

The influence of permanent magnet size parameters $a,b,h$ on sensing performance (sensitivity and ALR) is further analyzed by plotting a two-dimensional intensity map (Fig. 3d–f). Obviously $a$ is the most important parameter because it is related to the size of ALR. The relationship between $a$ and sensitivity is not monotonic, especially in the area where g is small (Fig. S2a in supplementary materials). But it is more important to note that the effect of the change of a on ALR is complex: there are some specific combinations of $a$ and g values that make ALR reach the maximum value (Fig. 3d). Under some clear design requirements, such as ALR = 5 mm, 10 mm or 20 mm, we can draw equipotential lines and further determine the values of a and g.

The increase of b will lead to a monotonous increase in sensitivity, because the increase of b is equivalent to placing multiple permanent magnets in parallel in the y-axis direction, significantly increasing the magnetic field strength at the location of the TMR sensor (Fig. S2b in supplementary materials). Unlike a, changes in b have little effect on ALR (Fig. 3e): it is unwise to try to improve ALR by increasing b when the system deployment size is limited.

Finally, it is evident from equation (9) that the larger the value of h, the higher the sensitivity, regardless of the value of g. This is also corroborated by the simulations shown in Fig. S2c, d in supplementary materials. The former and the latter show the change of sensitivity when a = 40 mm and a = 20 mm. They are all monotonic, because the increase of h can also be regarded as multiple permanent magnets connected in parallel. Regarding ALR, there is also a band-shaped maximum area (Fig. 3f), and the increase of a will cause the band area to move right (Fig. S2e for a = 20 mm and Fig. S2f for a = 40 mm in supplementary materials). Outside the band area, the increase of h will only slightly increase ALR.

In summary, the following conclusions have been drawn:

1. The parameters a, b, and ℎ of the permanent magnet contribute to the improvement of sensitivity to some extent. For instance, larger values of b and ℎ result in increased sensitivity, and there exists an optimal value of a that maximizes sensitivity. However, the primary determining factor is g: the larger the g, the lower the sensitivity.

2. The ALR is primarily determined by a and g, with b and ℎ having a lesser impact. There are some suitable combinations of a and g that make the ALR at a maximum value. Through a large number of simulation calculations, the suitable g is about half of a.

3. For the step of determining system parameters, first of all, it is necessary to determine the values of a and g according to the requirements of sensitivity and ALR. When sensitivity is more prioritized, smaller a and g are often selected, and when ALR is more prioritized, larger a and g are often selected. Secondly, select appropriate b and ℎ based on the determined a and g. Since the optimal values of b and ℎ are in a band area, it is more appropriate to select smaller b and ℎ when the sensitivity requirement is not high, which can greatly reduce the size of the entire system.

System configuration and integration

According to the Chinese national standard for displacement monitoring of buildings and bridge structures³⁶ and existing SHM research^37–39, we determined that the displacement monitoring system designed in this work should have a range of 15 mm and a resolution better than 0.1 mm. After the above analysis, we finally determined the parameters of the permanent magnet as shown in Table 1. At this time, under the control of 1% MEC and 0.5% ACE, a maximum ALR of 25.35 mm can be obtained when g is 19.35 mm (Fig. S3a in supplementary materials). Among all TMR commercial devices, BMM350 is a strong candidate due to its low power consumption and high resolution. It has an excellent temperature behavior with an outstanding low temperature coefficient (Table 2). When the relative displacement between the TMR sensor and the permanent magnet is within ±7.5 mm, the measured $B_x$ range is within ±1951 μT (Fig. S3b in supplementary materials). Therefore, the sensitivity of magnetic intensity-displacement can be calculated as 260.13 μT/mm. Considering the resolution of BMM350 is 0.1 μT, the highest resolution of the proposed displacement measurement method can reach 0.4 μm, and the zero offset drift does not exceed ±8 μm.

Table 1.

Parameter values of permanent magnet

Parameter	Values
Component	SmCo
$B_{\mathrm{r}}$	0.91 T
$a$	40 mm
$b$	8 mm
$h$	4 mm
g	20 mm

Table 2.

Parameters of TMR component

Parameter	Values
Model specification	BMM350
Range	±2 mT
Zero offset drift	±2 μT
Resolution	0.1 μT
Typical power	0.36 mW

In order to achieve long-term, autonomous, and intelligent monitoring of displacement under unmanned conditions, the selected TMR sensor is integrated into a microsystem with other functions such as processing, communication, and power supply (Fig. 4a). We used a three-dimensional vertical stacking interconnection approach to complete the integration of the microsystem (Fig. S4 in supplementary materials): each functional module was individually processed into a layer of high-density circuit, and different circuit layers were vertically stacked and electrically interconnected on the side walls. All functions of the microsystem are highly customizable, and substrates with required functions can be easily added according to actual monitoring needs. For example, we process the TMR sensor into a submodule separately, which can be easily stacked with any substrate. When the microsystem integrates a substrate with other SHM sensors, the TMR submodule can complete the electrical connection in the reserved holes (Fig. S5 in supplementary materials). In addition, we designed a mobile application that can wirelessly receive and visualize displacement data in real time (Fig. 4b).

Fig. 4. Overall configuration of the system.

a Photo of the microsystem with scale and RMB 1 coin as size comparison. b Designed mobile terminal visualization application. c Assembly of microsystem with TMR sensor, permanent magnet and sliding guide. d Schematic diagram of sensitivity self-calibration steps. e The guide groove limits the displacement range to 15 mm for sensitivity self-calibration

A guide rail system was designed to effectively convert the cracks or slips in the structure into the relative displacement between the TMR sensor and the permanent magnet. As shown in Fig. 4c, the microsystem and the permanent magnet are fixed on the base and slider of the guide rail, respectively, and reasonable grooves are reserved on them for fixing and adjusting the spacing g. When the position of the system changes in the magnetic field, the TMR sensor measures the change in the magnetic field, and the microcontroller unit (MCU) can calculate the corresponding relative displacement in real time. Sliding guide is usually made of antimagnetic materials such as aluminum alloy or engineering-plastics. After simulation calculation, it is proved that the influence of guide material on displacement error is within ±3 μm (Fig. S6d in supplementary materials). In addition, it is also necessary to consider the error of installation in the y-axis direction. When the y-axis installation error is 0.1 mm, 0.2 mm, 0.5 mm and 1 mm respectively, the x-axis magnetic field error is 0.1–4.5 μT respectively (Fig. S6a, b in supplementary materials). The corresponding displacement measurement error generally does not exceed 10 μm. Only when the Y-axis installation position deviation reaches 1 mm, there will be a displacement measurement error of about 25 μm (Fig. S6c in supplementary materials).

Although the conversion sensitivity of magnetic field intensity and displacement has been obtained through theoretical calculation, in practical application, the magnetization intensity of the permanent magnet will be affected by the manufacturing process, ambient temperature and attenuation, so it is difficult to keep constant. On the other hand, there is also an installation error in the distance g between the permanent magnet and the TMR sensor, so the sensitivity needs to be adaptively adjusted (Fig. 4d). The designed guide rail has a groove to determine the linear working range of the system. Before the system is installed and fixed, the sensitivity calibration link can be entered: the permanent magnet base slider is moved to the two ends of the groove (Fig. 4e). The extreme value of the magnetic field intensity is recorded, and the corrected sensitivity is obtained according to the extreme value (Fig. S7 in supplementary materials). When the groove limit of $x$ is −7.5 mm, $B_x$ is −1882 ± 0.2 μT; when the groove limit of $x$ is + 7.5 mm, $B_x$ is 1863 ± 0.2 μT. Considering that the machining groove has a processing error of 0.05 mm, the sensitivity $S_x$ with the error range added is finally 249.7 ± 0.9 μT/mm.

Results and applications

We compared the proposed TMR-based high-precision displacement measurement system with the laser ranging method, which was acknowledged to have excellent displacement measurement performance. The test environment was set up as shown in Fig. 5a. The displacement was controlled by a laser (0.01 mm measurement accuracy) to be changed from −7.5 mm to +7.5 mm with a 1.5 mm interval, and the displacement value measured by the laser was taken as the true value. At least five measurements were made and the errors of the two methods were calculated (Fig. 5b). It can be seen that the error was controlled within ±60 μm, and the maximum relative error corresponding to the ±7.5 mm range was less than 1%, which did not exceed the 1% MEC specified in the previous parameter optimization process. When approaching the edge of the range, a larger error occurred, which may be related to the saturation of the TMR sensor in a strong magnetic environment, resulting in distortion of the measurement results. In addition, we also verified the stability of the system under long-term operation. The system was independently powered by lithium-ion batteries, and the TMR sensor readings were recorded at 5 second intervals after the system was continuously operated for one week (Fig. S8a in supplementary materials). The results show that the total standard deviation is 0.563 μT, which is 2.25 μm long-term working accuracy; the maximum deviation is 3.09 μT, corresponding to a maximum position deviation of 12.4 μm. After filtering the original signal to extract the trend line, the drift velocity of $B_x$ over time can be obtained (Fig. S9a in supplementary materials), and the maximum drift velocity does not exceed 0.6 μT/h. The proposed system was tested under the same conditions as the displacement monitoring system equipped with Hall-effect sensor and AMR sensor (detailed parameters see Table S1 in supplementary materials), and the long-term working stability and drift speed were analyzed (Figs. S8 and S9 in supplementary materials). It can be seen that the TMR sensor is superior to other types of magnetometers in terms of power consumption, noise standard deviation, displacement drift speed and maximum displacement deviation (Fig. 5c).

Fig. 5. Characterization of the performance of a high-precision displacement monitoring microsystem.

a Experimental setup for accuracy comparison with laser ranging. b Error in the proposed system displacement measurement result when the laser ranging result is used as the true displacement. c Comparison of power consumption, noise standard deviation, displacement drift speed and maximum displacement deviation of three magnetometers

Here, we demonstrate three typical applications. It is well known that the tensile strength of concrete structures is much lower than the compressive strength. In some special cases, concrete structures may be damaged due to tension in some parts. In engineering practice, fiber aggregates are often added to concrete to improve its tensile properties⁴⁰. The proposed system is used to monitor the deformation of concrete structures under constant displacement load. The experimental setup is shown in Fig. 6a. The fixture applies a constant displacement load of 40 μm/min to the carbon fiber tension plate, which is precast with the concrete structure through the fiber. The proposed displacement monitoring system was arranged on the tension plate and the concrete structure base to measure the relative displacement at a frequency of 1 Hz. The experiment was terminated when the tension plate and the concrete structure separated. From the deformation results (Fig. 6b), the concrete structure undergoes three stages of elastic deformation, pre-splitting (plastic deforestation) and fracture. After tearing, the displacement change rate is consistent with the constant displacement load.

Fig. 6. Application demonstration of high-precision displacement monitoring microsystem.

a Experimental device for deformation monitoring of concrete structure. b Measured deformation displacement of concrete structure under displacement load. c Experimental device for monitoring the change of structural crack width caused by settlement. d Crack width curve under periodic impact load and static load. e Experimental device for monitoring lateral slip of urban bridge bearings. f Slip curve of bridge bearing

The second application is to monitor the change in crack width in structures. The proposed displacement monitoring system is placed at both ends of the crack to continuously monitor the width change of the crack under load (Fig. 6c). First, a 200 N load was used for periodic impact, and the displacement measurement results showed that the crack width changed from −120 μm to +50 μm. When a 500 N static load was applied, the crack width decreased by 40 μm, and when a 700 N static load was applied, the crack width continued to decrease by 55 μm (Fig. 6d).

The last application is to monitor the slip deformation of the bridge beam end on the bearing. The principle of the bridge bearing is to reliably transmit the load and deformation (displacement and rotation) borne by the bridge superstructure to the bridge substructure by connecting the superstructure and substructure of the bridge. Unreasonable loads will cause abnormal displacement of the beam end and may further lead to accidents, so it is particularly important to monitor the high-precision slip change of the beam end. The proposed system is arranged on the bearing at the bottom of the urban expressway bridge (Fig. 6e). Here we made some modifications to the guide rail frame: aluminum profiles are directly used as the guide rail base, which is fixed on the bridge bearing together with the microsystem integrated with the TMR sensor; the permanent magnet is fixed on the slider, and the slider is connected to the beam end through a connecting rod. In about 2 min, a typical vehicle passing by will produce a weak displacement within ±20 μm (Fig. 6f), and a truck passing by will produce a larger displacement (peak-to-peak value exceeds 50 μm).

To validate the long-term autonomous performance of the proposed displacement monitoring microsystem in real bridge environments, it was deployed at the Zhengzhou-Xinxiang Yellow River Bridge—a dual-purpose highway and high-speed railway bridge in Henan Province, China. The encapsulated microsystem was installed at the expansion joint of the bridge pier to continuously monitor displacement variations over a 39-day winter period (Fig. 7a). Powered solely by harvested solar energy, the system operated reliably under harsh weather conditions, including rain, snow, and significant temperature fluctuations. An integrated temperature-humidity sensor concurrently recorded environmental variations during the test (Fig. 7b). Analysis of the displacement data revealed periodic daily fluctuations in the expansion joint width, attributable to thermal effects (Fig. 7c). By fitting the temperature and displacement variations (Fig. 7d), thermally induced displacements (Fig. 7e) were successfully decoupled from dynamic load-induced displacements (Fig. 7f). Two distinct dynamic displacement sources were identified: high-speed trains caused displacements ranging from 15 μm to 100 μm, while vehicular traffic generated displacements below 15 μm. The microsystem demonstrated wireless alerting capabilities for abnormal displacements exceeding normal thresholds. For instance, on day 4 of deployment, a displacement exceeding 150 μm was detected. After excluding human factors, if such abnormal values continue to occur and the frequency tends to increase, further on-site inspections is necessary to assess potential structural risks.

Fig. 7. Application of the displacement monitoring microsystem in a real bridge.

a Encapsulated microsystem deployed at the expansion joint of a large bridge pier. b Environmental temperature and humidity variations during the 39-day test period. c Raw displacement data of the expansion joint. d Scatter plot of displacement versus temperature and their fitted relationship. e Thermally induced displacements. f Dynamic load-induced displacements

Discussion and conclusion

Based on the TMR sensing effect, this study completed the modeling optimization of the linear region of the magnetic field displacement relationship, designed a vertically stacked integrated microsystem, and achieved breakthrough progress in high-precision structural displacement monitoring.

In this work, our important contribution is to establish an analytical model for the linear region of the magnetic field-displacement relationship and perform multi-parameter optimization. The optimization results show that under the control of MEC and AEC, there is a set of combined parameters that makes ALR reach the local optimum and obtains the final sensitivity reach 250 μT/mm and the range reach 15 mm performance.

From a microsystem design and integration standpoint, we implemented a modular, vertically stacked architecture, where each functional unit—TMR sensor, signal processing, wireless communication, and power management—is independently fabricated and interconnected via sidewall. To ensure adaptability in real-world environments, we introduced a sensitivity self-calibration method based on pre-installed guide grooves. This design accounts for manufacturing and installation variabilities by enabling in-situ calibration of the sensor displacement sensitivity. The size of the entire system is controlled at 20 mm × 16 mm × 12 mm (3.84 cm³), and the cost of core components does not exceed US$20.

Finally, through performance characterization, it was verified that the designed system resolution reached 0.4μm, the error was less than 1%, and the accuracy was 2.25 μm. Comparing this work with other state-of-the-art for structural displacement monitoring, it can be seen that the proposed system can achieve the optimal measurement performance while the cost, size, and power consumption can be much smaller than other solutions (Table 3). The size only includes the sensor header. The volume of the TMR sensor module in this work is only 20 mm³.

Table 3.

Comparison of this work with other state-of-the-art works

	Accuracy	Cost	Size	Typical power
This work	2.25 μm	~$20	4 mm × 5 mm × 1 mm	0.55 mW
GNSS41	3.5 mm	~$1000	284 mm × 140 mm × 37 mm	5.7 W
Fiber9	0.1 mm	~$25000	φ = 38.5 mm, L=180 mm	Wired power
Laser42	0.32 μm	~$25000	φ = 35 mm, L=280 mm	Wired power
Vision43	0.15 mm	~$2000	φ = 143 mm	8.3 W
Hall-effect38	11.5 μm	~$20	8 mm × 10 mm	50 mW

The system was successfully applied to multiple SHM scenarios, including concrete structure deformation monitoring, crack width variation tracking, and bridge beam end slip detection. Moreover, in a 39-day deployment on the Zhengzhou-Xinxiang Yellow River Bridge, the system operated autonomously in harsh environmental conditions, tracking thermally induced daily displacement fluctuations and identifying abnormal structural responses. The results demonstrated the system’s ability to distinguish between environmental and dynamic load effects, thereby improving interpretability and actionable decision-making for maintenance.

In summary, this work provides both theoretical foundations and empirical evidence for a scalable, high-precision TMR-based displacement monitoring system suitable for long-term SHM. The combination of a physically compact microsystem, robust signal calibration, and reliable wireless communication makes it highly adaptable for diverse infrastructure applications. Future work will explore multi-axis displacement monitoring and integration with other SHM modalities to further enhance structural diagnostics.

Author contributions

X.L. designed and fabricated a wireless displacement monitoring microsystem. X.L. and Q.W. analyzed the sensing principle and conducted performance tests. Z.F. and N.W. completed the application experiment and data analysis. W.L. and Z. Y. supervised this work and provided technical and theoretical support. All the authors reviewed and approved the paper.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflict of interest

The authors declare no conflict of interest.

Supplementary information

The online version contains supplementary material available at 10.1038/s41378-025-01090-3.