Performance analysis of passive wheel speed sensor by 3D finite element method

Abstract

Passive wheel speed sensors are widely used in automotive applications where knowledge of the rotational velocity of the rotating axes and resistant to environmental impacts are required. This paper presents a general introduction to wheel speed sensor systems and the underlying physical behaviour are explained with the help of three-dimensional time-stepping finite element method. The 3D field simulations well reflect the dynamics of the magnetic field of the real problem and they provide a quantitative understanding of the output signal behaviour concerning model parameter variations. The sensor geometry is analyzed in detail, featuring a study of the influence of the main parameters. As an outcome of parameter analysis, it was possible to determine the influence of the sensor parameters on the output signal of the sensor. In addition, this study aims to advance the understanding of the physical behaviour of passive wheel speed sensors by three-dimensional nonlinear finite element analysis, which is missing from the literature.

Keywords

Wheel speed sensor passive sensor parameter analysis 3D finite element method

1. Introduction

Nowadays automotive electronic systems will further improve the safety of transport, as well as comfort, engine efficiency and electrification of vehicles. The strongly increasing application of automotive systems will create a strong demand for robust, inexpensive and high-performance magnetic sensors to determine the vehicle velocity and to integrate into drive assistance systems [1]. However, as the number of applications grows, the specifications they are expected to meet more and more demanding. For instance, larger signal amplitudes are required to improve signal-to-noise ratios and to relax manufacturing tolerances and lowering the cost.

There are several principles to realize speed sensors including mechanical, optical and magnetic solutions. Magnetic technology such as eddy current, inductive, Hall effect or giant magnetoresistance (GMR) is widely used to measure the rotational speed of vehicles or even revolution or position of different kind of shafts, axels, toothed wheels in gears, to name a few [1–3]. The measurement principles of magnetic sensors make contact-free interactions possible, which show an excellent robustness against temperature and dirt without the need of airtight seals or other environmental contamination control. For cost reasons, the passive inductive sensors have been used in the transport industry as a part of intelligent automotive vehicle electronics [1]. In addition to the low price, advantages of passive wheel speed sensors are high resolution, require no external power source and robustness against dirt. With the advantages, there are serious disadvantages of an inductive sensor because the output signal depends on the air gap, the speed of the toothed wheel, remanent magnetization of magnet and material of the core and toothed target wheel [2,4].

A typical passive inductive or also called variable reluctance wheel speed sensor configuration is shown in Fig. 1. It consists of a permanent magnet, a soft magnetic core, toothed wheel and a coil. The passive speed sensor is excited by the magnetic field, that is periodically modulated when the toothed wheel rotates. The system responds to the rotational movement of wheel past its magnetic steel core end by inducing voltage in the pick-up coil. An approximately sinusoidal output is detected for continuous wheel movement and the frequency of the output signal is used for determining the speed of the wheel.

Fig. 1.

Schematic of a passive wheel sensor system with toothed wheel.

The two main design processes of variable reluctance wheel speed sensor are based on magnetic circuit and finite element method. The aim of analytic magnetic circuit approach is a fast and easy design process [5] to obtains an approximation about the properties of output signal [6]. The finite element simulations allow the sophisticated analysis of sensor behaviour without the necessity of fabricating and testing many prototypes. However, most of the time, the finite element simulation is two-dimensional static magnetic [7–9] with linear magnetic media because of the simplification of the simulation, so the effect of third-dimension and motion of wheel, eddy current effect, and nonlinearity are neglected. Lequesne et al. [10] used a special two-dimensional approximation, the weighted sum of results in XZ and XY planes for taking into account the effect of third-dimension. Two- and three-dimensional results of stationary finite element calculation with measurements have been compared in [11], and three-dimensional stationary FE simulations have been used for analysing the proposed sensor structure in [12]. The tangential and radial component of air-gap field of the sensor is also important in the numerical analysis of the output signal as a function of pitch position as investigated in [8,9]. In addition, the effects of sensor components on the impedance sensitivity are calculated by means of the experimental design method in [13].

However, the numerical analysis of passive magnetic speed sensors has not yet been sufficiently resolved. The effect of eddy current and nonlinearity of B-H relation, the third-dimension, the variations of the sensor parameters and the effect of toothed wheel velocity are so far overlooked in the literature. This paper focuses to take into account the above mentioned neglected effects and to clarify the relation of these effects on the output signal, the induced voltage through a sensitivity analysis. Further, the authors investigated the influence of the parameters of the speed sensor on the output signal to improve the understanding of such systems for manufacturers and to form a solid basis for future work on this topic. All of the simulations have been carried out at low speed because it is the critical operating range for this type of sensor.

2. Passive inductive sensor

The passive inductive or variable reluctance wheel speed sensor schematic configuration is shown in Fig. 1. It consists of a permanent magnet, a soft magnetic core, toothed target wheel and a pick-up coil. The sensor applies the stationary magnetic field generated by a permanent magnet, that is periodically modulated when the toothed wheel rotates. When the target wheel rotates, the teeth, the slots and the permanent magnet assume different positions, which can be characterised by the angular distance between some arbitrary location on the toothed wheel and some other arbitrary location on the stationary magnet. The system responds to the rotational movement of wheel past its magnetic core end by inducing voltage in the pick-up coil. An approximately sinusoidal voltage output is detected for continuous wheel movement and the frequency of output signal used for determining the speed of the wheel.

The parameters of the passive wheel speed sensors, that have a decisive influence on the output signal can be roughly grouped into three categories: the parameters that describe the toothed wheel, the parameters that describe the magnet and the remaining parameters that describe the geometry of core and distance between the wheel and core [8]. The focus of this work lies on the parameters of the magnet, the length of the core, the number of turns of pick-up coil and a key geometric parameter, the air gap, while keeping all others fix (e.g. magnet volume, geometry of toothed wheel) at typical or reasonable values.

The cylindrical magnet is fully described by its geometry and magnetization parameters, the coercive field strength H _c and remanent induction B _r. The cylindrical symmetric core is stepwise thinned from the magnet to the air gap which is surrounded by the receiver coil as you can see in Fig. 1.

Concerning material parameters, in all numerical simulation, isotropic material featuring nonlinear constitutive relation was used. The used nonlinear magnetization curves can be seen in Fig. 2. In some cases, the nonlinear B-H relation is linearized [9], because the magnetic field inside the sensor parts does not exceed several hundreds of millitesla. In this work, this is not true as you can see in Fig. 3. Figure 3 shows the magnetic flux density distribution in the cross-section of the sensor, where the peak value is more than 1.6 T and the magnetic field intensity in the cross-section of the coil. Therefore, consideration of nonlinearity is necessary, this is also supported by the relative permeability on the surface of the sensor core (see in Fig. 3).

Fig. 2.

B-H relationships of materials [14,15].

Fig. 3.

Magnetic flux density distribution and magnetic field intensity in the cross-section of core and coil, respectively for 2nd magnet (left). Relative permeability on the iron core and toothed wheel surfaces for 2nd magnet (right).

3. Computation method

The performance and sensitivity analysis of a wheel speed sensor design requires the precise computation of magnetic behaviour and induced voltage. The speed sensor geometry typically varies in all three spatial dimensions. Because of it, the computational method in this study was based on the computational intensive, but accurate three-dimensional time-stepping finite element method.

The simulations were performed with the help of the finite element package ANSYS Maxwell [15]. The geometry of the problem is discretized into tetrahedron elements. The used finite element magnetic field formulation is based on magnetodynamic Maxwell’s equations. The numerical solution of these equations is based on ungauged $\boldsymbol{T}, \Phi - \Phi$ – formulation [16–19], which is a very efficient formulation for solving three-dimensional eddy current field problems. The weak formulation of the used finite element formulation coupled with the voltage equation of the coil is the following [17,18] $\begin{eqnarray}\displaystyle & \displaystyle \int _{{\Omega}_{\text{C}}}{\displaystyle \frac{1}{{\sigma}}}({\nabla}\times \boldsymbol{W}_{i})\cdot ({\nabla}\times \boldsymbol{T})∼\text{d}{\Omega}+\int _{{\Omega}_{\text{C}}}{\mu}\boldsymbol{W}_{i}\cdot \left({\displaystyle \frac{\partial \boldsymbol{T}}{\partial t}}+{\nabla}{\displaystyle \frac{\partial {\Phi}}{\partial t}}+{\displaystyle \frac{\partial i_{\text{C}}\boldsymbol{T}_{\text{C}}}{\partial t}}\right)∼\text{d}{\Omega}=\int _{{\Omega}_{\text{C}}}{\mu}\boldsymbol{W}_{i}\cdot {\displaystyle \frac{\partial \boldsymbol{T}_{0}}{\partial t}}∼\text{d}{\Omega}, & \displaystyle\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle & \displaystyle \int _{{\Omega}_{\text{C}}}{\mu}{\nabla}N_{i}\cdot \boldsymbol{T}∼\text{d}{\Omega}+\int _{{\Omega}_{\text{N}}+{\Omega}_{\text{S}}}{\mu}{\nabla}N_{i}\cdot ({\nabla}{\Phi}+{\Sigma}i_{\text{C}}\boldsymbol{T}_{\text{C}})∼\text{d}{\Omega}=\int _{{\Omega}_{\text{N}}+{\Omega}_{\text{S}}}{\mu}{\nabla}N_{i}\cdot \boldsymbol{T}_{0}∼\text{d}{\Omega}, & \displaystyle\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle & \displaystyle \int _{{\Omega}_{\text{S}}}{\mu}\boldsymbol{T}_{\text{C}}\cdot \left({\displaystyle \frac{\partial \boldsymbol{T}}{\partial t}}+{\nabla}{\displaystyle \frac{\partial {\Phi}}{\partial t}}+{\displaystyle \frac{\partial i_{\text{C}}\boldsymbol{T}_{\text{C}}}{\partial t}}\right)\text{d}{\Omega}+R_{\text{C}}i_{\text{C}}+L_{\text{C}}{\displaystyle \frac{\text{d}i_{\text{C}}}{\text{d}t}}=v_{\text{C}}, & \displaystyle\end{eqnarray}$ (3) where $\textit{i}$ = 1, …, the number of finite elements, $\sigma$ and $\mu$ are the conductivity and the permeability, $\textit{R}_C$ and $\textit{L}_C$ are the resistance and inductance in series with voltage source $\textit{v}_C$ and $\textit{i}_C$ is the unknown current in the voltage-driven coil.

In our study, the problem is subdivided into three disjunct regions, $\Omega = \Omega_{\text{C}}\cup\Omega_{\text{N}}\cup\Omega_{\text{S}}$ , where $\Omega_{\text{C}}$ is the conducting region with eddy current (sensor core, magnet and toothed wheel), $\Omega_{\text{S}}$ is the coil region and $\Omega_{\text{N}}$ is the air region. In the used formulation, the magnetic scalar potential $\Phi$ is represented by a nodal-based shape function, $N$ in the entire domain, while current vector potential $\boldsymbol{T}$ is represented by edge-based shape function, $\boldsymbol{W}$ in the conducting regions [16–18]. The resulting nonlinear sparse equation system is solved using sparse Gaussian elimination direct solver with Newton-Raphson method [15,16].

4. Results and discussion

The variable parameters of the implemented model are the length of the core, the number of turns of pick-up coil and the air gap width. The shape of the pick-up coil is depending on the core length and the number of turns. All the other components and parameters keep fixing. The permanent magnet has also fixed size, only the material changes. Two different anisotropic cast magnets were tested, 1st magnet (B _r = 1. 24 T, H _c = 51600 A/m) and 2nd magnet (B _r = 1. 05 T, H _c = 119336 A/m) at the sensor [14]. The target wheel is made of cast steel and the ferromagnetic core is made of Steel 1008. The first magnetization curve of these materials of these materials is shown in Fig. 2.

All simulations have been studied using the same finite element mesh, which consists of 528263 tetrahedron elements and the number of unknowns is 729011. The moving band technique [16,17] has been used for the movement of the toothed wheel.

The results of the sensitivity analysis can be seen in Fig. 4. From the curves, it can be seen how the normalized magnitude of the induced voltage changes as a function of the varied parameters. Two additional parameters, the magnet radii and the rotation speed of wheel were tested in the sensitivity analysis. The magnet radii varied between 2 mm and 5 mm, the wheel velocity varied between 0.3 rps and 3.9 rps. The volume of the magnet is constant during the analysis. As shown in the figure, the air gap has the greatest impact on the output.

Fig. 4.

The sensitivity of output signal to parameters change.

The results of the simulation concerning the amplitude of induced voltage are shown in Figs 5, 6, 7 and 8. The wheel rotational speed is constant at 1 rotate per second (60 rpm) during all these simulations, as you can see in Fig. 3. The reason for this low rotation speed is that this is the critical range of variable reluctance sensors because the signal-to-noise ratio is bad in this speed range. These figures show the peak-to-peak value of induced voltage in the function of the number of conductors, core length and air gap. The number of conductors varied between 5000 and 10175, the core length varied between 7 mm and 22.5 mm and the air gap varied between 0.1 mm and 1.9 mm. However, this is the actual air gap between the sensor and the wheel. Magnetically, the air gap varies from 1.1 mm to 2.9 mm, because the non-magnetic housing has a thickness of 1 mm. The used parameter range is determined by the available space for the sensor. The air gap is 1 mm in the case of Figs 5 and 6, and the core length is 18.6 mm in the case of Figs 7 and 8. These air gap size and core length are common for these types of sensors.

Fig. 5.

Magnitude of the induced voltage in the function of core height and number of conductors for 1st magnet.

Fig. 6.

Magnitude of the induced voltage in the function of core height and number of conductors for 2nd magnet.

Fig. 7.

Magnitude of the induced voltage in the function of air gap and number of conductors for 1st magnet.

As expected the magnitude of induced voltage in the case of 1st magnet exceeds the voltage of the speed sensor with 2nd magnet, because 1st magnet has a higher remanent induction. The surface as shown in Fig. 6 is even more uniform as shown in Fig. 5 because higher B _r makes it easier to saturate the core. This is also supported by the fact that, with a smaller core length, the surface of the induced voltage is even more uneven. Nonetheless, the induced voltage roughly linearly depends on the length of the core and on the number of conductors. The relationship between induced voltage and the air gap is strongly nonlinear as shown in Figs 7 and 8. The local minima and local maxima are shown in Figs 5 and 6 and the nonlinear relationship between the air gap and the voltage supports the importance of taking the nonlinearity at the sensor material into consideration. In all four figures, around the maximum, the saturation can be observed, which roughly gives the maximum voltage available through the construction. The maximum peak-to-peak values of induced voltage in the case of 1st magnet and 2nd magnet are 969.5 mV and 777.4 mV, respectively.

Fig. 8.

Magnitude of the induced voltage in the function of air gap and number of conductors for 2nd magnet.

5. Conclusions

This paper presents a parametric analysis of a passive wheel speed sensor by the help of three-dimensional time-stepping nonlinear finite element method. The wheel speed sensor behaviour was performed by sensitivity and parametric analysis, where the main parameter was the output signal, the peak to peak value of induced voltage.

The results show that precise mounting is very important for this sensor because a small air gap defect significantly affects the output signal. There is an approximately linear relationship between the output signal and the other analyzed parameters, such as the number of conductors, the velocity of toothed wheel, the magnet radii and the core length. The local minima and local maxima and the nonlinear relationship between the parameters make the use of three-dimensional numerical simulation justified in the design of this type of sensor.

Our future plan is to examine the effect of the eddy current on the output signal and to take into account the phenomena arising from manufacturing technology and assembly. In addition, the magnetic losses of wheel speed sensor are analyzed in the function of speed.

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