Improved Design of an Eddy-Current Speed Sensor Based on Harmonic Modeling Technique

Abstract

This study proposes an improved design of an eddy-current speed sensor (ECSS) by adding a ferromagnetic core to the stator, resulting in a sensitivity enhancement ranging from three to sixty times compared to a reference model according to shaft materials. An improved analytical model (AM) based on harmonic modeling (HM) is developed to account for the effects of core permeability, validated through finite element analysis (FEA), demonstrating excellent agreement between the two methods. Based on this model, the optimal dimensions of the proposed design are obtained, and comprehensive analyses of shaft materials and excitation source parameters are performed. The results show that the magnetic shaft offers the highest sensitivity, while a nonmagnetic shaft with low conductivity ensures optimal linearity. Meanwhile, a nonmagnetic shaft with high conductivity leads to low sensitivity and higher linearity errors. Furthermore, a high-frequency excitation source enhances output linearity but necessitates careful selection based on the shaft materials. The dynamic characteristics of the proposed design under different operating conditions are analyzed using a coupled Ansys Twin Builder and Maxwell 2D model. The proposed design and AM significantly improve ECSS performance and the analyzing tool, providing a robust and practical solution for precise speed measurement in various applications.

1. Introduction

Over the past few decades, permanent magnet synchronous machines (PMSMs) have been extensively developed and are now the backbone of various applications due to their high efficiency, power density, and precise control capabilities [1,2]. However, induction machines (IMs) remain widely used in industry, accounting for approximately 80% of industrial applications [3]. This preference is largely due to their low production costs, simple structure and control, and high durability under extreme temperatures and harsh environments. To ensure the optimal performance of IMs under varying load conditions, accurate and reliable speed measurement is crucial.

Sensorless estimation techniques have been widely studied because of their non-intrusive and contactless structure. Nevertheless, these methods face significant challenges, such as complex hardware requirements for signal processing, susceptibility to electrical faults in stator windings, and slow response times under dynamic conditions [4]. Conventional measurements rely on devices such as DC tachometers [5,6], optical encoders, variable reluctance sensors [7,8,9,10], and resolvers [11,12,13,14] that need a connection to the shaft. DC tachometers, although widely used, incur high maintenance costs due to their brush structure, while optical encoders are particularly sensitive to environmental factors such as dust and dirt. Variable reluctance sensors and resolvers are cost effective, reliable, and robust but require complicated signal processing [15].

Recently, eddy-current speed sensors (ECSSs) have emerged as a novel solution with contactless and non-intrusive approaches. Overall, there are two types, categorized based on the excitation source. The first type [4,15,16,17,18,19,20] is excited by a high-frequency source (HFS), and the other [21,22,23,24] operates using a permanent magnet (PM) source. Evidently, the latter is simpler and more compact, but it also has drawbacks. Low-frequency or static excitation fields can introduce interference errors in the Hall effect [15], resulting in a limited linear range, while the PM is further susceptible to temperature changes. Although the first type requires a wave generator for excitation, it mitigates temperature effects due to the stable magnetic field maintained by a constant excitation current. Furthermore, with HFS, the first offers higher accuracy, with a maximum linearity deviation of only 0.15% compared to 1.89% of the PM type in the same 3000 rpm testing range, as shown in [17,24], respectively. Hence, the operating range can be extended beyond that of the PM type. These types of ECSSs feature several topologies applied to various objects, such as conductive surfaces [16,18], solid conductive shafts [17,19,20], and hollow conductive shafts [15]. These sensors can achieve excellent linearity, with errors smaller than 0.15% [15], a wide detection range, and minimal impact from shaft eccentricity [17]. However, these sensors generally have low sensitivity, often producing signals in the range of several µV/rpm. As a result, signal amplification is necessary, which not only increases the cost of signal processing but also makes the system more vulnerable to interference from external magnetic fields.

Additionally, the design, optimization, and analysis of ECSS require computational tools such as finite element analysis (FEA) or analytical models (AMs). FEA, whether in 2D or 3D, offers significant advantages in terms of high accuracy and the ability to model complex geometries. Nevertheless, its primary limitation lies in the long computation time, especially when addressing dominant eddy-current effects, which involve intricate interactions between magnetic fields and conductive materials. In contrast, AMs are often preferred for simpler problems due to their computational efficiency. Many AMs have been developed based on the subdomain method (SDM) for ECSS [17,19,20] and electrical machines, considering both magnetostatic [25,26,27] and time-dependent problems [28,29]. Despite the maturity, the limitations of SDM in accounting for core permeability restrict its applications. Fortunately, the harmonic modeling (HM) technique has recently been proposed as a novel approach to consider permeability. Since the first research [30] was introduced in 2016, many extended studies have been conducted to consider permeability, saturation effects [31,32,33,34], and irreversible demagnetization [35]. However, most of these studies are performed under magnetostatic assumptions, neglecting the time dependency of the problem, which prevents them from being applied to the ECSS analysis.

This paper proposes an improved design based on the ECSS introduced in [17] by integrating a low-reluctance core into the stator, resulting in a sensitivity increase of three to sixty times compared to a reference model [17]. Furthermore, an improved analytical model based on HM that considers core permeability is presented to account for the effects of the low-reluctance core accurately. Validation through FEA demonstrates the exceptional accuracy of the proposed AM. Steady-state analyses performed by the AM are used to evaluate the effects of shaft material and excitation-source parameters on the design performance and determine the optimal dimensions of the proposed design.

Furthermore, dynamic characteristics of the improved ECSS, which were not extensively addressed in previous studies [15,17,19,20], are captured using a coupled Ansys Twinbuilder and Maxwell 2D model. This coupled approach allows for a comprehensive evaluation of sensor performance under varying operating conditions, highlighting the advantages of the proposed design in both steady-state and dynamic scenarios.

2. Proposed Design and Operation Principle

This study is based on an ECSS described in [17], with the excitation and pick-up coils positioned perpendicularly, as shown in Figure 1a. The induced eddy currents in conductive objects consist of two components: the transformer component resulting from the time variation in the source field and the motional component arising from the relative movement between the conductive object and the excitation field [36]. The transformer component produces a symmetric flux distribution in the conductive object, whereas the motional component introduces asymmetry [37,38], which leads to a net flux through the pick-up coil that is greater than zero, thereby inducing a voltage. Obviously, there are two ways to increase this voltage: increasing the motional speed of the object or strengthening the excitation field. Since the speed cannot be controlled in ECSS, the proposed design enhances the excitation field strength by incorporating low-reluctance paths, such as teeth and yoke, effectively guiding the magnetic flux, as depicted in Figure 1b. A detailed 3D representation of the new structure is shown in Figure 1c. This improved structure improves the sensor’s sensitivity and provides a robust frame to support the windings.

The operational principle of the proposed design is depicted in Figure 2. A high-frequency current is injected into the excitation coils to generate a stable magnetic field. At zero speed, Figure 2a illustrates the symmetric magnetic flux distribution generated solely by the transformer component, resulting in zero flux linkage in the pick-up coil. Meanwhile, when the shaft rotates, the motional component creates an asymmetric flux distribution, as shown in Figure 2b, which induces a voltage in the pick-up coils. This induced voltage is proportional to the shaft speed, making the sensor highly suitable for accurate speed measurement.

Figure 2c compares the output voltage waveforms of the proposed and reference designs [17] for different shaft materials: nonmagnetic shafts (aluminum and stainless steel) and a magnetic shaft (cast iron). The excitation source is set to an amplitude of 150 mA, with the shaft speed at 3000 rpm. Across all tested frequencies, the proposed design consistently produces a significantly higher output voltage than the reference design. Specifically, the minimum and maximum increases are 3.1 and 61.8 times at 750 Hz, 3.3 and 76.3 times at 500 Hz, and 3.8 and 104.2 times at 250 Hz. Therefore, these results demonstrate that the proposed design substantially improves ECSS performance while maintaining the same power consumption.

3. 2D Analytical Model Using Harmonic Modeling

The analytical model is performed in the 2D cylindrical coordinate system, and the symbols utilized in this section are shown in Table 1. As depicted in Figure 1b, the sensor is divided into five regions: (I) shaft, (II) air gap, (III) excitation coil region, (IV) pick-up coil region, and (V) yoke. Due to the small excitation source used in the sensor, saturation effects in the ferromagnetic parts can be neglected. Therefore, permeability in the ferromagnetic parts is considered uniform, denoted as ${\mu }_{st}$, and to simplify the problem further, the following assumptions are made:

• In each region, permeability is variant in tangential and invariant in radial directions.

• The end effects are neglected.

3.1. Complex Fourier Series Representation

The magnetic vector potential, magnetic flux density, magnetic field strength, and current density distribution can be expressed in terms of the complex Fourier series (CFS), which considers time dependency $t$ as follows.

(1)$\overrightarrow{A}=A_z\left(r,\theta ,t\right){\overrightarrow{u}}_z$

(2)$\overrightarrow{B}=B_r\left(r,\theta ,t\right){\overrightarrow{u}}_r+B_{\theta }\left(r,\theta ,t\right){\overrightarrow{u}}_{\theta }$

(3)$\overrightarrow{H}=H_r\left(r,\theta ,t\right){\overrightarrow{u}}_r+H_{\theta }\left(r,\theta ,t\right){\overrightarrow{u}}_{\theta }$

where ${\overrightarrow{u}}_z,{\overrightarrow{u}}_r,{\overrightarrow{u}}_{\theta }$ are unit vectors, and expressions $X\left(r,\theta ,t\right)$ in Equations (1)–(3) are given by.

(4)$X\left(r,\theta ,t\right)=\sum _{k=-\infty }^{\infty }{\displaystyle \sum _{n=-\infty }^{\infty }}{\widehat{X}}_{k,n}\left(r\right)e^{j\left(k{\omega }_{s}t-n\theta \right)}$

where $k$ is the time-harmonic order, $n$ is the spatial-harmonic order, and ${\omega }_s=2\pi f_s$ is the time-based rotational speed of the excitation field.

Assuming that the sensor is excited by a single harmonic, Equation (4) is simplified as.

(5)$X\left(r,\theta ,t\right)=\sum _{n=-\infty }^{\infty }{\widehat{X}}_n\left(r\right)e^{j\left({\omega }_{s}t-n\theta \right)}$

For practical calculations, the finite Fourier series are utilized.

(6)$X\left(r,\theta ,t\right)=\mathfrak{R}\left\{\sum _{n=-N}^N{\widehat{X}}_n\left(r\right)e^{j\left({\omega }_{s}t-n\theta \right)}\right\}$

where $N$ is the highest harmonic order considered.

Equation (6) is rewritten in the matrix form as follows.

(7)$X\left(r,\theta ,t\right)=\mathfrak{R}\left\{e^{j{\omega }_{s}t}{\mathit{X}}^T.\mathit{E}\right\}$

(8)$\mathit{X}={\left[{\widehat{X}}_{-N}\dots {\widehat{X}}_N\right]}^T; \mathit{E}={\left[e^{jN\theta }\dots 1\dots e^{-jN\theta }\right]}^T$

Therefore, the CFS coefficient vectors of electromagnetic quantities can be written as ${\mathit{A}}_z,{\mathit{B}}_r,{\mathit{B}}_{\theta },{\mathit{H}}_r,{\mathit{H}}_{\theta }$, and ${\mathit{J}}_z$, respectively. Additionally, permeability distribution in each region only varies in the tangential direction and is given by $\mu \left(\theta \right)=\mathit{\mu }$.

Cauchy’s product theorem and Fourier factorization method are used to express the relationship between the field strength $\mathit{H}$ and flux density $\mathit{B}$ as follows [31].

(9)${\mathit{B}}_r={\mathit{\mu }}_{c,r}{\mathit{H}}_r; {\mathit{B}}_{\theta }={\mathit{\mu }}_{c,\theta }{\mathit{H}}_{\theta }$

where ${\mathit{\mu }}_{c,r}, {\mathit{\mu }}_{c,\theta }$ are the convolution matrices of permeability Fourier series coefficients on the radial and tangential directions, respectively, expressed as

(10)${\mathit{\mu }}_{c,r}=\left[\begin{array}{ccc}{\widehat{\mu }}_0& \cdots & {\widehat{\mu }}_{-2N}\\ ⋮& \ddots & ⋮\\ {\widehat{\mu }}_{2N}& \cdots & {\widehat{\mu }}_0\end{array}\right]$

(11)${\mathit{\mu }}_{c,\theta }=\left[\begin{array}{ccc}{\widehat{\mu }}_0^{rec}& \cdots & {\widehat{\mu }}_{-2N}^{rec}\\ ⋮& \ddots & ⋮\\ {\widehat{\mu }}_{2N}^{rec}& \cdots & {\widehat{\mu }}_0^{rec}\end{array}\right]$

3.2. Excitation Source and Permeability Convolution Matrix

Figure 3a shows the current density distribution in Region (III), and CFS coefficients ${\widehat{J}}_{z,n}$ are given by.

(12)${\widehat{J}}_{z,n}=\left\{\begin{array}{c}0;n=0\\ {\displaystyle \frac{J_0}{\pi n}}\left[\sin \left({\displaystyle \frac{n\left(\pi -{\theta }_{ei}\right)}{2}}\right)+\sin \left({\displaystyle \frac{n\left(\pi +{\theta }_{\mathit{ei}}\right)}{2}}\right)-\sin \left({\displaystyle \frac{n\left(\pi -{\theta }_{eo}\right)}{2}}\right)-\sin \left({\displaystyle \frac{n\left(\pi +{\theta }_{eo}\right)}{2}}\right)\right]\\ ;n\ne 0\end{array}\right.$

where $J_0={\displaystyle \frac{N_t^{e}I\left(t\right)}{\left({\theta }_{eo}-{\theta }_{ei}\right)\left(R_{eo}^2-R_{ei}^2\right)/4}}$; $N_t^e$ is the number of turns of the excitation coils.

From the permeability distribution in Regions (III) and (IV) described in Figure 3b, the CFS coefficients of permeability in Region (III) are written as

(13)${\widehat{\mu }}_0^{III}={\displaystyle \frac{2{\mu }_{st}\left({\alpha }_1+{\alpha }_2\right)+{\mu }_0\left({\gamma }_2+2{\gamma }_1\right)}{\pi }}; n=0$

(14)$\begin{array}{l}{\widehat{\mu }}_n^{III}={\displaystyle \frac{1}{n\pi }}\sum _{i=1}^2{e}^{jn{\theta }_{t1}^i}\left[{\mu }_0\sin n\left({\alpha }_1+{\gamma }_1\right)+\left({\mu }_{st}-{\mu }_0\right)\sin \left(n{\alpha }_1\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+e^{jn{\theta }_{t2}^i}\left[{\mu }_{st}\sin n\left({\displaystyle \frac{{\gamma }_2}{2}}+{\alpha }_2\right)+\left({\mu }_0-{\mu }_{st}\right)\sin \left({\displaystyle \frac{n{\gamma }_2}{2}}\right)\right];n\ne 0\end{array}$

For Region (IV):

(15)${\widehat{\mu }}_0^{IV}={\displaystyle \frac{{\mu }_{st}\left({\alpha }_1^{’}+{\gamma }_2^{’}\right)+2{\mu }_0{\gamma }_1^{’}}{\pi }} ;n=0$

(16)${\widehat{\mu }}_n^{IV}={\displaystyle \frac{1}{n\pi }}\sum _{i=1}^2{e}^{jn{\theta }_{t1}^i}\left[{\mu }_0\sin n\left({\gamma }_1^{’}+{\displaystyle \frac{{\gamma }_2^{’}}{2}}\right)+\left({\mu }_{st}-{\mu }_0\right)\sin \left(n{\displaystyle \frac{{\gamma }_2^{’}}{2}}\right)\right]+e^{jn{\theta }_{t2}^i}{\mu }_{st}\sin n{\displaystyle \frac{{\alpha }_1^{’}}{2}};n\ne 0$

where ${\theta }_{t1}^i=\left(i-1\right)\pi$ and ${\theta }_{t2}^i=\left(2i-1\right)\pi /2$.

The inverse coefficient ${\widehat{\mu }}_n^{rec}$ in Equation (11) is obtained by replacing ${\mu }_0$ and ${\mu }_{st}$ with their inverse value ${1/\mu }_0$, ${1/\mu }_{st}$ in Equations (13)–(16).

3.3. Solution Derivation of Partial Differential Equations (PDE)

The magnetic field is derived from Ampere’s law as follows:

(17)$\nabla \times \overrightarrow{H}=\overrightarrow{J}$

(18)${\displaystyle \frac{\partial H_{\theta }}{\partial r}}+{\displaystyle \frac{H_{\theta }}{r}}-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial H_r}{\partial \theta }}=J_{ext}-\sigma \left({\displaystyle \frac{\partial A}{\partial t}}+{\omega }_r{\displaystyle \frac{\partial A}{\partial \theta }}\right)$

where $J_{ext}$ is the external excitation source, such as current density, and the second component on the right-hand side represents the induced eddy current in the conductive object resulting from both temporal and spatial variation. $\sigma , {\omega }_r$ are the conductivity and the electric angular speed of the moving object.

From Equations (6) and (7), the components of Equation (18) are developed as follows:

(19)${\displaystyle \frac{\partial H_{\theta }}{\partial r}}=e^{j{\omega }_{s}t}{\displaystyle \frac{\partial {\mathit{H}}_{\theta }^T}{\partial r}}\mathit{E};{\displaystyle \frac{\partial H_r}{\partial \theta }}=-j{e}^{j{\omega }_{s}t}{\mathit{H}}_r^T{\mathit{K}}_{\theta }\mathit{E}$

(20)$J_{ext}=e^{j{\omega }_{s}t}{\mathit{J}}_{ext}^T\mathit{E}$

(21)${\displaystyle \frac{\partial A}{\partial t}}=j{\omega }_s{e}^{j{\omega }_{s}t}{\mathit{A}}_{\mathit{z}}^T\mathit{E}; {\displaystyle \frac{\partial A}{\partial \theta }}=-j{e}^{j{\omega }_{s}t}{\mathit{A}}_{\mathit{z}}^T{\mathit{K}}_{\theta }\mathit{E}$

where ${\mathit{K}}_{\theta }$ is the diagonal matrix of $N$, ${\mathit{K}}_{\theta }=diag\left(-N\dots N\right)$.

Using Equations (19)–(21), Equation (18) can be expressed without the time and spatial dependency as follows.

(22)${\displaystyle \frac{\partial {\mathit{H}}_{\theta }}{\partial r}}+{\displaystyle \frac{{\mathit{H}}_{\theta }}{r}}+j{\displaystyle \frac{1}{r}}{\mathit{K}}_{\theta }{\mathit{H}}_r={\mathit{J}}_{ext}-j\sigma \left({\omega }_s-{\omega }_r{\mathit{K}}_{\theta }\right){\mathit{A}}_z$

The relationship between flux density and magnetic vector potential is written as.

(23)$\overrightarrow{B}=\nabla \times \overrightarrow{A}$

(24)$B_r={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial A_z}{\partial \theta }};B_{\theta }=-{\displaystyle \frac{\partial A_z}{\partial r}}$

In matrix form, Equation (24) is given by

(25)${\mathit{B}}_r=-j{\displaystyle \frac{1}{r}}{e}^{j{\omega }_{s}t}{\mathit{K}}_{\theta }{\mathit{A}}_z;{\mathit{B}}_{\theta }=-{\displaystyle \frac{\partial {\mathit{A}}_z}{\partial r}}$

Replace Equation (24) into Equation (9), ${\mathit{H}}_r$ and ${\mathit{H}}_{\theta }$ are obtained as

(26)${\mathit{H}}_r=-j{\displaystyle \frac{1}{r}}{\mathit{\mu }}_{c,r}^{-1}{\mathit{K}}_{\theta }{\mathit{A}}_z;{\mathit{H}}_{\theta }=-{\mathit{\mu }}_{c,\theta }^{-1}{\displaystyle \frac{\partial {\mathit{A}}_z}{\partial r}}$

Substituting Equation (26) into Equation (22), the final partial differential equation is derived.

(27)${\displaystyle \frac{{\partial }^2{\mathit{A}}_z}{\partial r^2}}+{\displaystyle \frac{1}{r}} {\displaystyle \frac{\partial {\mathit{A}}_z}{\partial r}}-{\displaystyle \frac{1}{r^2}}{\mathit{V}\mathit{A}}_z=-{\mathit{\mu }}_{c,\theta }\left[{\mathit{J}}_{ext}-j\sigma \left({\omega }_s-{\omega }_r{\mathit{K}}_{\theta }\right){\mathit{A}}_z\right]$

where $\mathit{V}={\mathit{\mu }}_{c,\theta }{\mathit{K}}_{\theta }{\mathit{\mu }}_{c,r}^{-1}{\mathit{K}}_{\theta }$.

From Equation (27), the PDEs for each Region (I)–(V) are listed in Equations (28)–(30), respectively.

(28)${\displaystyle \frac{{\partial }^2{\mathit{A}}_z}{\partial r^2}}+{\displaystyle \frac{1}{r}} {\displaystyle \frac{\partial {\mathit{A}}_z}{\partial r}}-{\displaystyle \frac{1}{r^2}}{\mathit{V}\mathit{A}}_z=j{\sigma }_{sh}{\mu }_{sh}\left({\omega }_s-{\omega }_r{\mathit{K}}_{\theta }\right){\mathit{A}}_z;\mathrm{f}\mathrm{o}\mathrm{r}\left(\mathrm{I}\right)$

(29)${\displaystyle \frac{{\partial }^2{\mathit{A}}_z}{\partial r^2}}+{\displaystyle \frac{1}{r}} {\displaystyle \frac{\partial {\mathit{A}}_z}{\partial r}}-{\displaystyle \frac{1}{r^2}}{\mathit{V}\mathit{A}}_z=0;\mathrm{f}\mathrm{o}\mathrm{r}\left(\mathrm{I}\mathrm{I}\right),\left(\mathrm{I}\mathrm{V}\right),\left(\mathrm{V}\right)$

(30)${\displaystyle \frac{{\partial }^2{\mathit{A}}_z}{\partial r^2}}+{\displaystyle \frac{1}{r}} {\displaystyle \frac{\partial {\mathit{A}}_z}{\partial r}}-{\displaystyle \frac{1}{r^2}}{\mathit{V}\mathit{A}}_z={\mathit{\mu }}_{c,\theta }{\mathit{J}}_{ext};\mathrm{f}\mathrm{o}\mathrm{r}\left(\mathrm{I}\mathrm{I}\mathrm{I}\right)$

where ${\sigma }_{sh}$ and ${\mu }_{sh}$ are the conductivity and permeability of the shaft material.

Using the method of separation of variables to solve Equations (28)–(30), the corresponding solutions are given by

(31)${\mathit{A}}_z^k=I_{\mathit{T}}\left(\mathit{\alpha }r\right){\mathit{a}}^k+K_{\mathit{T}}\left(\mathit{\alpha }r\right){\mathit{b}}^k;k=\left(\mathrm{I}\right)$

(32)${\mathit{A}}_{\mathit{z}}^k={\left({\displaystyle \frac{r}{R_{outer}}}\right)}^{{\mathit{\lambda }}^k}{\mathit{a}}^k+{\left({\displaystyle \frac{r}{R_{inner}}}\right)}^{-{\mathit{\lambda }}^k }{\mathit{b}}^k;k=\left(\mathrm{I}\mathrm{I}\right),\left(\mathrm{I}\mathrm{V}\right),\left(\mathrm{V}\right)$

(33)${\mathit{A}}_{\mathit{z}}^k={\mathit{W}}^k{\left({\displaystyle \frac{r}{R_{eo}}}\right)}^{{\mathit{\lambda }}^k}{\mathit{a}}^k+{\mathit{W}}^k{\left({\displaystyle \frac{r}{R_{ei}}}\right)}^{-{\mathit{\lambda }}^k}{\mathit{b}}^k+r^2{\mathit{F}}^k;k=\left(\mathrm{I}\mathrm{I}\mathrm{I}\right)$

where $I_{\mathit{T}}$ and $K_{\mathit{T}}$ are the first and second kinds of Bessel function with orders of $\mathit{T}=\left|{\mathit{K}}_{\theta }\right|$, $\mathit{\alpha }={\left(j{\sigma }_{sh}{\mu }_{sh}\left({\omega }_s-{\omega }_r{\mathit{K}}_{\theta }\right)\right)}^{0.5}$. $R_{outer}$ and $R_{inner}$ are the outer and inner radii of the $k$-th regions. ${\mathit{W}}^k,{\mathit{\lambda }}^k$ are the eigenvector matrix and diagonal eigenvalue of ${\left({\mathit{V}}^k\right)}^{0.5}$. ${\mathit{F}}^k$ is the particular solution of Equation (30), given by

(34)${\mathit{F}}^k={\left({\mathit{V}}^k-4\mathit{I}\right)}^{-1}{\mathit{J}}_{ext}$

3.4. Boundary Conditions (BCs)

To obtain the electromagnetic field in the sensor, boundary conditions (BCs) on the continuity of the radial flux density component ${\mathit{B}}_r$ and tangential field strength ${\mathit{H}}_r$ are applied to four interfaces at $R_r$, $R_{ei}$, $R_{eo}$, and $R_{po}$. Moreover, Dirichlet BCs are used for both the outer boundary of the Region (V) and the infinitesimal radius of the Region (I). The solution is derived by solving the following ten conditions.

(35)${\mathit{A}}_{\mathit{z}}^I{|}_{r\to 0}=0; {\mathit{A}}_{\mathit{z}}^V{|}_{r=R_y}=0$

(36)${\mathit{A}}_{\mathit{z}}^I{|}_{r=R_r}-{\mathit{A}}_{\mathit{z}}^{II}{|}_{r=R_r}=0; {\mathit{H}}_{\mathit{z}}^I{|}_{r=R_r}-{\mathit{H}}_{\mathit{z}}^{II}{|}_{r=R_r}=0$

(37)${\mathit{A}}_{\mathit{z}}^{II}{|}_{r=R_{ei}}-{\mathit{A}}_{\mathit{z}}^{III}{|}_{r=R_{ei}}=0; {\mathit{H}}_{\mathit{z}}^{II}{|}_{r=R_{ei}}-{\mathit{H}}_{\mathit{z}}^{III}{|}_{r=R_{ei}}=0$

(38)${\mathit{A}}_{\mathit{z}}^{III}{|}_{r=R_{eo}}-{\mathit{A}}_{\mathit{z}}^{IV}{|}_{r=R_{eo}}=0; {\mathit{H}}_{\mathit{z}}^{III}{|}_{r=R_{eo}}-{\mathit{H}}_{\mathit{z}}^{IV}{|}_{r=R_{eo}}=0$

(39)${\mathit{A}}_{\mathit{z}}^{IV}{|}_{r=R_{po}}-{\mathit{A}}_{\mathit{z}}^V{|}_{r=R_{po}}=0; {\mathit{H}}_{\mathit{z}}^{IV}{|}_{r=R_{po}}-{\mathit{H}}_{\mathit{z}}^V{|}_{r=R_{po}}=0$

These ten BCs are used to obtain a linear system $\mathit{A}\mathit{X}=\mathit{B}$ to determine all unknown coefficients $\mathit{X}$ of magnetic vector potential in all regions. This system is implemented in MATLAB R2023b, in which $\mathit{A}$ is the square matrix of the coefficient factor, and $\mathit{B}$ is a column vector of the constant values with the size of $10\left(2N+1\right)$ and $\left(2N+1\right)$, respectively. Increasing the harmonic order N enhances the solution’s accuracy but also increases computation time. To address the zero-harmonic issue in the truncated CFS, a small value, such as 10⁻¹⁰, is assigned to the zero harmonic instead of setting it to zero [32].

3.5. Derivation of Pick-Up Voltage and Excitation Coil Inductance

The pick-up voltage is a crucial parameter of the sensor. The flux passing through one solid conductor $i$ of the pick-up coil is expressed by

(40)${\phi }_i^p={\displaystyle \frac{L_{stk}}{S_p^{con}}}{\int }_{{\theta }_p^i}^{{\theta }_p^i+{\displaystyle \frac{{\theta }_{po}-{\theta }_{pi}}{2}}}{\int }_{R_{pi}}^{R_{po}}{A}_z^{IV}(r,\theta )rdrd\theta$

where $S_p^{con}=\left({\theta }_{po}-{\theta }_{pi}\right)\left(R_{po}^2-R_{pi}^2\right)/4$ is the area of a solid conductor.

From the winding connection of the pick-up coil, the total flux linkage is given by

(41)${\psi }^p=N_t^p\left({\phi }_1^p+{\phi }_2^p-{\phi }_3^p-{\phi }_4^p\right)$

where $N_t^p$ is the number of turns in the pick-up coil.

Finally, the induced voltage in the pick-up coil is derived.

(42)$E_p=-{\displaystyle \frac{d{\psi }^{IV}}{dt}}$

To find the inductance of the excitation coil $L_e$, similar steps are used to obtain total flux linkage ${\psi }^e$ first, and then the excitation coil inductance is derived by.

(43)$L_e={\displaystyle \frac{{\psi }^e}{I}}$

where $I$ is the exciting current.

4. Analytical Model Verification by FEA

To verify the accuracy of the analytical model, the results obtained from AM and 2D time-transient FEA are compared. The current supplied to the excitation coils is set to 750 Hz and 150 mA. The main parameters of the proposed design and the shaft material properties are listed in Table 2 and Table 3, respectively.

At 3000 rpm, the flux density distributions in the air gap are shown in Figure 4a,b for two shaft materials: aluminum and cast iron. The flux density produced with the cast-iron shaft is significantly higher than that with the aluminum shaft. This is caused by the higher permeability of the cast iron $\left({\mu }_r=60\right)$, resulting in low reluctance paths for the flux flow. In contrast, with the aluminum shaft, the permeabilities of aluminum and air are equal $\left(\mu =1\right)$, leading to a very high reluctance in the rotor and minor flux density.

When the shaft speed is set at 40,000 rpm, the rotation frequency approaches the stator frequency, and eddy effects are diminished. As a result, the flux density in the air gap increases for both shaft materials, as shown in Figure 4c,d. Additionally, at the higher speed, a more significant tail effect [23] is generated in the aluminum shaft, leading to substantial growth of the tangential component. Notably, the AM results show excellent agreement with the FEA results, confirming the model’s validity.

For a clearer perspective, the relationship between pick-up voltage (root-mean-square value) and speed is illustrated in Figure 5. A linear relationship between pick-up voltage and speed is observed for both materials, with cast iron producing a much greater output due to its high permeability, as shown in Figure 5a. Moreover, as shown in Figure 5b, the AM predictions agree well with FEA, and the highest differences are around 0.8%.

While 2D models are less accurate than 3D calculations, 3D transient FEA is computationally intensive and time consuming [17]. Consequently, the AM is used to analyze the steady-state characteristics of the sensor in the following section.

5. Steady-State Analysis

5.1. Effects of Stator Dimensions on Sensitivity Coefficient and Excitation Coil Inductance

The sensitivity coefficient (SC: the ratio between pick-up voltage and rotation speed) and the excitation coil inductance (ECI) are two critical parameters of the ECSS. Higher sensitivity enhances the signal-to-noise ratio and increases resistance to electromagnetic interference, while lower ECI enables quicker response times by reducing the required stabilization time for the excitation current. To study the variations in SC and ECI with respect to stator dimensions, the width of the teeth in front of the pick-up coil is defined as $K_{sp1}={\alpha }_1/{\beta }_1$, and for the teeth in front of the excitation coil, the ratio is $K_{sp2}={\alpha }_2/{\beta }_2$, as shown in Figure 1b. At 3000 rpm and with an excitation current of 750 Hz and 150 mA, SC and ECI are shown in Figure 6 for varying $K_{sp1}$ and $K_{sp2}$.

In Figure 6c,e, SC is proportional to $K_{sp2}$ for cast-iron and stainless-steel shafts, whereas SC shows a dependency on $K_{sp1}$ for the aluminum shaft, as depicted in Figure 6a. The reasons are explained as follows.

• At 750 Hz current source, the penetration depths $\delta =1/\sqrt{\pi f\sigma {\mu }_0{\mu }_r}$ for aluminum, cast iron, and stainless steel are 3.92 mm, 1.94 mm, and 15.0 mm, respectively. This value represents the depth where the magnetic field strength decreases significantly.

• For the aluminum shaft, the small penetration depth leads to pronounced skin effects, concentrating the magnetic field near the surface of the shaft and significantly weakening the field in the air gap despite the larger excitation pole, as shown in Figure 7a. However, SC increases with the width of the pickup pole due to the greater flux linkage obtained.

• For the cast-iron shaft, SC increases notably with the excitation pole, primarily due to the low reluctance in the rotor and stator, which facilitates higher magnetic flux density (MFD). Although cast iron has the smallest penetration depth, the larger excitation pole induces a higher MFD in the air gap (see Figure 7b), thereby enhancing sensitivity.

• For the stainless-steel shaft, the considerable penetration depth minimizes skin effects, flux penetration is deep, and flux attenuation is negligible. Consequently, the extent of the excitation pole is directly proportional to SC, which increases due to the higher MFD in the air gap (see Figure 7c).

Regarding ECI, because the permeabilities of aluminum and stainless steel are similar to that of air, their ECI variations are identical and significantly smaller than those observed with the cast-iron shaft, which has much greater permeability (Figure 6b,d,f). Generally, the ECI increases with $K_{sp1}$ and $K_{sp2}$ due to the low reluctance along the magnetic path.

Despite the increased SC achieved by enlarging the teeth widths, the ECI also rises because wider teeth increase the magnetic path length and inductance. Consequently, a trade-off between SC and ECI is made by selecting the optimal values of $K_{sp1}=0.5$ and $K_{sp2}=0.4$. These dimensions are adopted for the analyses in the subsequent sections.

5.2. Effects of Shaft Material and Excitation Frequency on Sensitivity Coefficient and Linearity

The effects of shaft material and excitation frequency on the SC of the sensor are shown in Figure 8. First, in all cases, SC is inversely proportional to the conductivity of the shaft because higher conductivity intensifies eddy effects. This trend is similarly observed with increasing frequency from 250 Hz to 750 Hz, further exacerbating eddy effects.

Shaft permeability generally enhances SC by lowering magnetic reluctance, facilitating greater flux linkage. However, SC is not strictly proportional to permeability. As shown in Figure 8a,b, excessively high permeability can lead to magnetic saturation or serious skin effects, with the highest output occurring at a permeability of roughly $20{\mu }_0$. Therefore, the optimal SC can be achieved without relying on excessively high-permeability materials, which would increase the inductance of the excitation coil.

In [17], the pick-up coils are connected to a lock-in amplifier to extract the DC components of the induced voltage, as depicted in Figure 9. Furthermore, [17] indicates that the real component of the induced voltage, $U_r$, exhibits higher sensitivity and lower linearity errors than the imaginary counterpart, $U_i$. Consequently, this study focuses on extracting $U_r$ to evaluate the sensitivity and linearity of the proposed design.

Figure 10 shows the real components $U_r$ and linearity errors as speed functions under the same excitation-current amplitude of 150 mA. The cast-iron shaft induces the highest outputs $U_r$ because of the high permeability. Additionally, for aluminum and cast-iron shafts in Figure 10a,b, $U_r$ increases as the excitation frequency decreases, attributed to the reduction in eddy effects. Meanwhile, with the stainless-steel shaft in Figure 10c, where eddy-current effects are negligible, the output $U_r$ grows with increasing excitation frequency.

Regarding linearity error, this value is calculated based on the best-fitted linear line, $f_t\left(n\right)$, of the output $U_r$ according to speed, as defined by the following equation.

(44)$\epsilon =\left|{\displaystyle \frac{f_t\left(n\right)-U_r\left(n\right)}{f_t\left(n\right)}}\right|.100\%$

As shown in Figure 10d–f, the aluminum shaft exhibits the highest nonlinearity across all frequencies, with a maximum error of 10.7% at 250 Hz. Conversely, the stainless-steel shaft demonstrates excellent linearity characteristics, with a maximum error of 0.47%, the lowest among all materials, at 250 Hz. For the cast-iron shaft, the nonlinearity is relatively low, with a maximum error of 2.14%. It is noteworthy that increasing the excitation frequency improves the linearity of the sensor. Compared to [17], using the same 750 Hz excitation frequency and an aluminum shaft, the sensitivity increases by a factor of 3.1, despite the linearity error rising from 0.2% to 0.67%. However, this linearity is still relatively low compared to a commercial magnetic speed sensor with a 1% linearity error [17] and another ECSS configuration [19] with 0.5%.

6. Dynamic-State Analysis

To investigate the dynamic characteristics of the eddy-current speed sensor, the circuit in Figure 9 is implemented in Ansys Twinbuilder, coupled with Maxwell 2D. A reference speed is applied to the shaft, and the estimated speed is calculated as the ratio of the real component $U_r$ to the sensitivity coefficient (V/rpm). To mitigate the delay effects of the low-pass filter (LPF), it is replaced by an averaging block, which computes the average value of $U_r$ over each cycle, $1/f$.

The performance of the speed sensor is shown in Figure 11. First, as depicted in Figure 11d–f, the estimation error is significantly higher at low speeds due to reduced signal strength and increased noise interference. Since the induced voltage frequency is twice the excitation frequency, the response time is directly influenced by the excitation frequency. Consequently, increasing the excitation frequency from 250 Hz to 750 Hz significantly improves the sensor’s accuracy across all tested shaft materials, with estimation errors during the T = 0.1–0.8 s period reduced to less than 0.8%.

Second, comparing performance across shaft materials, the aluminum shaft paired with a 250 Hz excitation frequency shows the highest estimation errors, primarily due to the material’s high conductivity and low permeability, with errors exceeding 6% during T period, as shown in Figure 11d. In contrast, the stainless-steel shaft demonstrates the best dynamic performance, with estimation errors remaining below 0.8% during the T period, regardless of the excitation frequency (250 Hz or 750 Hz; see Figure 11f). The performance of the cast-iron shaft lies between that of the aluminum and stainless-steel shafts.

In conclusion, using a shaft material with low conductivity and permeability, such as stainless steel, provides the best estimation performance, as its low conductivity and permeability minimize eddy-current effects and improve flux linkage stability.

7. Conclusions

This study developed an improved design of an eddy-current speed sensor (ECSS) to overcome the low sensitivity problem. The sensitivity of the proposed design is up to 104 times greater than that of the reference model without increasing power consumption. This is achieved by adding a ferromagnetic core to the stator, strengthening the signal while keeping the system efficient.

An improved analytical model based on harmonic modeling was introduced to better account for the effects of core permeability. Using this model, the effects of shaft materials and excitation frequency on the sensor’s performance were analyzed in addition to obtaining the optimal sensor dimensions. The results indicate that cast iron exhibits the highest sensitivity due to its high permeability. In contrast, stainless steel provides the best linearity because of its low conductivity and reduced eddy effects.

Furthermore, this study also investigates the dynamic behavior of the sensor using coupled simulations in Ansys, which has been largely ignored in most studies. Analyzing the excitation frequency shows that higher frequencies improve dynamic precision but may reduce output strength due to stronger eddy currents.

These findings offer valuable insights into ECSS behaviors under different conditions. Future research should focus on experimental validation of the model.

8. Discussion

Although many studies have been conducted on eddy-current speed sensors (ECSSs), several critical research areas still require further investigation. While 3D FEA models provide the most accurate results, they are very time consuming. To address this, semi-analytical 3D models are needed to reduce computation time while maintaining accuracy. The authors of [15] proposed an analytical model that includes end effects for ECSSs using the subdomain method. However, as mentioned earlier, this model does not account for core permeability.

Low excitation frequencies can raise linearity errors, but connecting the sensor directly to the electrical grid to eliminate external waveform generators presents a promising solution to simplify ECSS systems. This configuration could reduce system complexity and cost and improve reliability. Future studies could explore combining different materials in a shaft for this purpose. As shown in [19], a rotating iron shaft with a copper coating can enhance both sensitivity and linearity.

Finally, thanks to the simple design of ECSS, particularly without permanent magnet excitation, it is well suited for high-temperature environments. However, the effects of temperature changes on ECSS performance still need to be explored. Temperature fluctuations can affect the permeability and conductivity of materials, leading to potential drift in sensor measurements.

Author Contributions

Conceptualization, D.-T.H.; methodology, D.-T.H.; validation, D.-T.H. and M.-D.N.; formal analysis, D.-T.H.; investigation, D.-T.H.; writing—original draft, D.-T.H.; writing—review and editing, Y.-J.K., A.-T.P., K.-H.S. and J.-Y.C.; visualization, D.-T.H.; supervision, Y.-J.K., A.-T.P., K.-H.S. and J.-Y.C. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. (a) Reference design [17]; (b) proposed design; and (c) 3D model of the proposed design.

Figure 2. Analytical results under a 750 Hz, 150 mA excitation source: flux distribution of the proposed design with the aluminum shaft at (a) zero speed, (b) 20,000 rpm, and (c) pick-up voltage comparison between the reference [17] and proposed design with different shafts at 3000 rpm.

Figure 3. (a) Current density distribution in Region (III). (b) Permeability distribution in Region (III) and Region (IV).

Figure 4. Flux density distribution in the air gap with different shaft materials at (a,b) 3000 rpm and (c,d) 40,000 rpm under the excitation current of 750 Hz, 150 mA.

Figure 5. (a) Pick-up coil voltage and (b) difference between FEA and AM results according to speed with different shaft materials under the excitation current of 750 Hz, 150 mA.

Figure 6. Sensitivity coefficient (Vrms/rpm) and excitation coil inductance (ECI) according to the teeth width ratios at 3000 rpm and an excitation of 750 Hz, 150 mA with shaft material of (a,b) aluminum, (c,d) cast iron, and (e,f) stainless steel.

Figure 6. Sensitivity coefficient (Vrms/rpm) and excitation coil inductance (ECI) according to the teeth width ratios at 3000 rpm and an excitation of 750 Hz, 150 mA with shaft material of (a,b) aluminum, (c,d) cast iron, and (e,f) stainless steel.

Figure 7. Magnetic flux density distribution (absolute value) at the air gap with two [Forumla omitted. See PDF.] values, (a) aluminum shaft, (b) cast iron shaft, and (c) stainless steel shaft.

Figure 8. Sensitivity coefficient (Vrms/rpm) variations according to the relative permeability and conductivity of the shaft with an excitation frequency of (a) 250 Hz, (b) 500 Hz, and (c) 750 Hz.

Figure 9. Schematic circuit model to extract DC components: pick up coils connected to the lock-in amplifier [17].

Figure 10. (a–c) Real components of output voltage and (d–f) linearity error according to speed with different shaft materials.

Figure 11. (a–c) Reference speed and the estimated speeds, and (d–f) estimation errors.

Table 1

Nomenclature.

Symbol	Explanation	Symbol	Explanation
$r/\theta /z$	Radius/tangential and axial directions	$\sigma$	Conductivity
$j$	Imaginary unit	${\omega }_r$	Electric angular velocity of moving object
$\overrightarrow{A}$	Magnetic vector potential	$\mathit{A}$	Matrix form of magnetic vector potential
$\overrightarrow{B}$	Magnetic flux density	$\mathit{B}$	Matrix form of magnetic flux density
$\overrightarrow{H}$	Magnetic field strength	$\mathit{H}$	Matrix form of magnetic field strength
$\overrightarrow{J_z}$	Current density vector	$\mathit{J}$	Matrix form of current density
$\mu$	Permeability	${\mathit{J}}_{ext}$	Matrix form of external current density
${\widehat{\mu }}_n$	CFS coefficient of nth harmonic	${\mathit{A}}^T$	$\mathrm{Transpose} \mathrm{matrix} \mathrm{of} \mathrm{matrix} \mathit{A}$
${\widehat{\mu }}_n^{rec}$	Inverse coefficient of nth harmonic	${\mathit{\mu }}_{c,r}$	Permeability convolution matrix in radial direction
${\mu }_0$	Vacuum permeability	${\mathit{\mu }}_{c,\theta }$	Permeability convolution matrix in tangential direction
${\mu }_{sh}$	Shaft permeability	${\mathit{K}}_{\theta }$	$\mathrm{Diagonal} \mathrm{matrix} \mathrm{of} N$
${\mu }_{st}$	Stator permeability	$I_{\mathit{T}}$	First kind of Bessel function
${\widehat{J}}_{z,n}$	CFS coefficient of nth harmonic	$K_{\mathit{T}}$	Second kind of Bessel function
$J_{ext}$	External excitation source	$N_t^p$	Number of turns in the pick-up coil
$k,n$	Time and spatial harmonic orders	$N_t^e$	Number of turns in the excitation coil
${\omega }_s,f_s$	Time-based velocity and frequency	$L_{stk}$	Stack length of the sensor
$\mathfrak{R}$	Real part	${\phi }_i^p$	$\mathrm{Flux} \mathrm{passing} \mathrm{through} \mathrm{a} \mathrm{conductor} i$ of the pick-up coil
${\psi }^p$	Flux linkage through the pick-up coil	${\psi }^e$	Flux linkage through the excitation coil
$N$	Highest harmonic order	$I$	Excitation current

Table 2

Proposed design parameters.

Parameter	Symbol	Unit	Value
Shaft radius	$R_r$	mm	15.0
Inner and outer radius of the excitation coil region	$R_{ei}/R_{eo}$	mm	16.2/19.0
Inner and outer radius of the pick-up coil region	$R_{pi}/R_{po}$	mm	19.3/22.2
Stator yoke radius	$R_y$	mm	24.2
Inner and outer span angle of the excitation coil	${\theta }_{ei}/{\theta }_{eo}$	deg.	127.5/175.1
Inner and outer span angle of the pick-up coil	${\theta }_{pi}/{\theta }_{po}$	deg.	135.2/175.8
Stack length	$L_{stk}$	mm	39.0
Number of turns of the excitation coil	$N_t^e$	-	90
Number of turns of the pick-up coil	$N_t^p$	-	105

Table 3

Shaft material parameters.

Parameter	Symbol	Unit	Value
Cast-aluminum conductivity	$\sigma$	S/m	22 × 106
Cast-iron conductivity	$\sigma$	S/m	15 × 105
Stainless-steel conductivity	$\sigma$	S/m	15 × 105
Relative cast-aluminum permeability	${\mu }_r$	-	1.0
Relative cast-iron permeability	${\mu }_r$	-	60.0
Relative stainless-steel permeability	${\mu }_r$	-	1.0

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