Calculation of faults between core laminations and prediction of axial damage depth of large motors

Abstract

A method to analytically calculate the parameters of the fault area between core laminations are proposed and the dynamic trend of the fault development is predicted to comprehensively and accurately obtain the fault degree and damage level of short-circuits between the core laminations of large motors. Considering the influences of frequency and magnetic flux on the axial skin depth of the core and of the eddy current loss on the insulation resistance between the laminations and reviewing the laminations of the stator core tooth as the research object, based on the Maxwell equations and equivalent circuit theory, analytic expressions of active power and induced voltage in the fault area between the laminations are derived; the influence of the number of fault laminations on the electrical parameters in the fault area is quantitatively analyzed; the actual fault development process of the short-circuit faults between the laminations are obtained; and the correctness and effectiveness of the proposed method are verified via numerical simulations and the experiment of faults between core laminations. The sequence of induced voltage changing with the number of fault laminations is selected as the grey sequence to establish the grey prediction model (GM (1,1)). The development curve of the damage depth of the faults between core laminations along the axial direction is illustrated by calculating the sequence data. The results show that the error between the predicted value of the damage depth of the short-circuit fault herein and the experimental value is less than 7%. This method can be used in engineering practices.

Keywords

Faults between core laminations analytic method eddy loss damage depth GM (1,1)

1. Introduction

The safe service and effective perception of faults have become increasingly important with the rapid development of large motors. In particular, the dynamic trend analysis of faults for the components of large high-voltage motors and structural design optimization is vital to ensure safety. A statistical analysis of accidents in large-scale power equipment has demonstrated that accidents caused by insulation failure attribute over 50% [1–3] of the total number of accidents, among which short-circuit accidents between the core laminations attribute a high proportion in all types of accidents. In serious cases, silicon lamination may be burned out. Figure 1 shows a burned stator core. Therefore, the calculation and analysis of the fault area between the core laminations and the prediction of the dynamic development trend are crucial in assessing the safe and stable operation of the motor.

Fig. 1.

A burned stator core.

The current analysis and calculation of faults between the core laminations mainly include the methods of analogy and homogenization, which perform the analysis from two aspects: the finite element calculation and the experimental detection. In [4–6], the simulation methods of the fault between the core laminations are proposed, which include a short-circuit winding coil around the core, inserting a copper strip between the laminations, and local welding, to artificially create a fault current circuit. In [7], the AC excitation system and the sensor probe are combined to determine whether an inter-plate fault exists in the detection area by collecting the change signal of the alternating magnetic field in the insulation area of the addendum. However, the aforementioned experimental testing methods require disassembling to be conducted under the condition that the motor is stopped, which is not only destructive to the performance characteristics of the motor but also difficult to conduct. In [8–10], based on the method of homogenization and equivalent circuit theory, the continuum of equivalent conductivity and permeabilitypermeability is used instead of the actual lamination to simulate the short-circuit fault, and the conductivity and permeabilityconductivitypermeability tensors in the fault area are calculated. The eddy current loss and eddy current loss density in the fault area are then analyzed and calculated using the finite element method. Although the calculation results are relatively accurate, a very detailed subdivision of each silicon lamination is required, which leads to a huge calculation cost and problems, such as large calculation scale, time consumption, and difficulty during application in engineering practice. When calculating the influence of the axial eddy current, the aforementioned research results did not consider the thermal influence of the eddy current on the insulation layer and the change in the resistance value of the insulation layer between the core laminations in the case of the short-circuit fault. The inter-plate short-circuit fault is a cumulative process, which is different from mechanical damage, such as fracture and stretching, and is a dynamic and gradual development process. Some scholars have established an equivalent circuit of silicon lamination that only considers the resistance to solve the eddy current loss in the fault area using the analytic method; they also simulated the dynamic process of inter-plate faults by changing the resistance value [11–15], but ignored the influence of inductance on these inter-plate faults.

Considering the practical core modeling, efficiency of grid generation, and dynamic development of the short-circuit fault, based on the analytic method and equivalent circuit theory and fully considering the influences of frequency and magnetic flux on the axial skin depth of the core, the equivalent circuit of the fault with resistance and inductance parameters is established herein; the analytic formulas of the magnetic field and the active and reactive power in the silicon lamination are derived in a three-dimensional (3D) rectangular coordinate system; moreover, the induced voltage in silicon lamination is derived. The correctness of the analytical algorithm is verified by the finite element simulation and the fault experiment between the laminations. Moreover, the sequence of the fault voltage change with the number of fault laminations is selected as the grey sequence to establish the grey GM (1,1) model and predict the axial damage depth and the development trend of the faults between the laminations and to draw the dynamic development curve of the faults between the core laminations to provide data reference for fault diagnosis and structural design optimization of cores in the future.

2. Calculation of the electrical parameters in the fault area based on the analytic method

The long-term effect of the alternating magnetic field, eddy current, and heat loss on the insulation between the core laminations has led to adhesiveness in the insulating material to be carbonized and ionized in the air to produce corrosive gas and a large number of high-speed charged particles, which will damage the surface structure of the mica plate and result in the reduction of the effective insulation thickness Δ between the core laminations. When it is reduced to a limit value of Δ_min, an electrical connection occurs at a certain point in the core lamination, and the eddy current between the laminations suddenly increases. At the same time, additional loss and heat are generated, resulting in a decrease in the insulation performance of the inter-plate insulation and an increase in the deterioration of the insulation, thus forming a short-circuit fault between the core laminations.

For the convenience of research, the short-circuit fault between the core laminations is divided into two phases: initial fault and complete fault. The coated insulation on both sides of the core plate is partially damaged when the initial fault occurs, causing an electrical connection between the lamination and the damaged insulation to form the fault current. The loss and heat generated by the fault current further destroys the insulation between the core laminations, expands the fault level, which eventually leads to the complete damage of the insulation layer in the fault area, and enters the complete fault phase.

2.1. Calculation of electrical parameters in the fault area

Theoretically, the inter-plate short-circuit fault may occur at any position of the core, but it can be roughly divided into two initial cases in the actual working condition: (1) an insulation fault exists on one side of the core, and the magnetic flux of the core induces the eddy current electromotive force at the fault location and generates the fault current, which flows in the fault area and returns along the positioning rebar, as shown in Fig. 2(a); and (2) the insulating varnish on both sides of the tooth is damaged; hence, the induced eddy current flows through the fault point to form a fault current path, as shown in Fig. 2(b). In these two cases, the insulating varnish still exists between the silicon laminations in the fault area, and the current between the silicon laminations flows through the damaged positions of the insulating varnish on both sides or the positioning rebar.

In this study, the case of the insulation damage on both sides of the stator tooth (Fig. 2(b)) is considered as an example to analyze the short-circuit fault between the core laminations. The damage is simulated by local welding. Therefore, the insulation resistance between the laminations in the fault area is replaced by the resistance R _con at the welding point [12,16]; $\begin{eqnarray}\displaystyle R_{\text{con}}={\displaystyle \frac{L_{w}}{{\sigma}\cdot s_{w}}} & & \displaystyle \nonumber\end{eqnarray}$ Where σ is the electrical conductivity of silicon lamination, L _w and S _w are the length and the sectional area, of the welding points in the path direction through which the fault current flows, respectively.

Fig. 2.

The insulation damage.

Considering the rated operation of the motor, the value of iron loss at the tooth area of the stator is relatively large, and the top of the tooth is susceptible to mechanical damage. Thus, performance degradation is likely to occur between the core laminations. Therefore, the stator tooth is selected as the research object. In order to simplify the calculation, the stator tooth section is regarded as a rectangle; the axial length of the tooth is l; the tooth width is b _t; and the groove depth is h _s, as shown in Fig. 3(a). The thickness of one piece of silicon lamination at the stator tooth is denoted as d. The conductivityconductivity is denoted by σ, and the permeabilitypermeability is μ. The angular frequency of the magnetic field is ω; the silicon lamination surface is parallel to the YOZ plane; and the origin o of the coordinate system is located in the center of the lamination, as shown in Fig. 3(b).

The following logical hypotheses are made to simplify the analysis and calculation of the electromagnetic field in one single piece of silicon lamination:

To simplify the calculation of the lamination resistance, the following assumptions have been made:

1. Under the condition of fault and non-fault, the variation law of magnetic flux of iron core is the same;

2. The magnetic field distribution in each silicon lamination is the same;

3. There are no quality defects, such as burrs or bumps on the surface of the silicon steel sheet;

4. The surface magnetic field intensity of the silicon lamination is averaged, and the surface magnetic field intensity at each position is approximately the same; therefore, on two planes parallel to YOZ, a magnetic field intensity with a magnitude of H ₀ exists in the Z direction;

5. The thickness d of one single piece of silicon lamination is far less than the tooth width, that is, d ≪ b _t.

Fig. 3.

Schematic diagram of stator core tooth.

Figure 3(b) shows that the equivalent resistance of one single piece of silicon lamination can be divided into two directions along the x- and y-axes, denoted as R ₁ and R ₂, respectively. R ₁ and R ₂ can be calculated using the following formulas, $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}R_{1}={\displaystyle \frac{1}{{\sigma}}}\cdot {\displaystyle \frac{d}{{\delta}\cdot h_{s}}}\\ R_{2}={\displaystyle \frac{1}{{\sigma}}}\cdot {\displaystyle \frac{b_{t}}{d\cdot h_{s}}}\end{array}\right. & & \displaystyle\end{eqnarray}$ (1) Where δ is skin depth, ${\delta}=\sqrt{\frac{2}{{\omega}{\mu}{\sigma}}}$ .

Respectively: d ≪ b _t, then R ₁ ≪ R ₂; hence, R ₁ is negligible, and the equivalent circuit diagram of one single piece of silicon lamination is similar to that shown in Fig. 4, where E is the induced voltage of one single piece of silicon lamination, and L ₂ is the inductance of one single piece of silicon lamination.

Fig. 4.

The equivalent circuit diagram of single silicon steel sheet.

The fault domain equivalent circuit in the initial fault phase is shown in Fig. 5.

Fig. 5.

The initial fault area equivalent circuit.

In the phase of complete fault, the insulation paint between the laminations is considered to be nonexistent, and the equivalent circuit of fault area is shown in Fig. 6.

Fig. 6.

The equivalent circuit diagram of the complete fault area.

2.2. Calculation of active and reactive power in silicon lamination

The silicon lamination in the time-varying magnetic field induces the eddy current, and the magnetic flux in the silicon lamination is unevenly distributed because of the reaction of the eddy current, thereby generating a skin effect in the magnetic field of the silicon lamination. Hypothesis (4) shows that the magnetic field intensity in the silicon lamination is only the component ${\dot{H}}_{\text{z}}$ in the Z direction, and ${\dot{H}}_{\text{z}}$ is only a function of coordinate X; thus, ${\dot{H}}_{\text{z}}$ satisfies the one-dimensional complex eddy current equation [17]. $\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial ^{2}\overset{\cdot }{H}_{z}}{\partial x^{2}}}=j{\omega}{\sigma}{\mu}\overset{\cdot }{H}_{z} & & \displaystyle\end{eqnarray}$ (2) Where $p=\sqrt{j{\omega}{\sigma}{\mu}}=\frac{1+j}{{\delta}}$ .

According to ((2)), the test solution of ${\dot{H}}_{z}$ . $\begin{eqnarray}\displaystyle \overset{\cdot }{H}_{z}=A\cosh (px)+B\sinh (px) & & \displaystyle\end{eqnarray}$ (3) Where A, B is the unknown quantity to be solved, and they are determined by boundary conditions.

The boundary condition of magnetic field in silicon steel sheet is $\begin{eqnarray}\displaystyle x=\pm {\displaystyle \frac{d}{2}},\quad \overset{\cdot }{H}_{z}=H_{0}. & & \displaystyle \nonumber\end{eqnarray}$

According to the boundary conditions, the general solution of Eq. (3) can be deduced as follows $\begin{eqnarray}\displaystyle \overset{\cdot }{H}_{z}=H_{0}{\displaystyle \frac{\cosh (px)}{\cosh \left(p{\displaystyle \frac{d}{2}}\right)}}. & & \displaystyle\end{eqnarray}$ (4)

Since ${\dot{H}}_{z}$ does not change with y, it is only a function of coordinate X, so its current density only has Y component. According to (4), the current density is $\begin{eqnarray}\displaystyle \overset{\cdot }{J}_{y}=-{\displaystyle \frac{\partial \overset{\cdot }{H}_{z}}{\partial x}}=-H_{0}p{\displaystyle \frac{\sinh (px)}{\cosh \left(p{\displaystyle \frac{d}{2}}\right)}}. & & \displaystyle\end{eqnarray}$ (5)

At the surface $x=\frac{d}{2}$ of silicon steel sheet, if the electric density is $\dot{J}_{0}$ , it can be seen from (5) $\begin{eqnarray}\displaystyle \overset{\cdot }{J}_{0}=\overset{\cdot }{J}_{y\left(x={\displaystyle \frac{d}{2}}\right)}=-H_{0}p\cdot \text{th}\left(p{\displaystyle \frac{d}{2}}\right). & & \displaystyle\end{eqnarray}$ (6)

Similarly, at the surface $x=-\frac{d}{2}$ of silicon steel sheet, $\dot{J}_{y}=-\dot{J}_{0}$ .

Simultaneous (5) and (6), the electric field strength ${\dot{E}}_{y}$ can be seen from (7) $\begin{eqnarray}\displaystyle \overset{\cdot }{E}_{y}={\displaystyle \frac{1}{{\sigma}}}\overset{\cdot }{J}_{y}=\overset{\cdot }{E}_{0}{\displaystyle \frac{\sinh (px)}{\sinh \left(p{\displaystyle \frac{d}{2}}\right)}} & & \displaystyle\end{eqnarray}$ (7) where ${\dot{E}}_{0}=\frac{1}{{\sigma}}\dot{J}_{0}$ .

According to (5), (6) and (7), the complex energy flow vector of silicon steel sheet surface is $\begin{eqnarray}\displaystyle \overset{\cdot }{\prod }_{\left(x=\pm {\displaystyle \frac{d}{2}}\right)}=\pm {\displaystyle \frac{1}{2}}\overset{\cdot }{E}_{0}H_{0}=\mp {\displaystyle \frac{1}{2}}H_{0}^{2}{\displaystyle \frac{1+j}{{\sigma}{\delta}}}\text{th}\left[(1+j){\displaystyle \frac{d}{2{\delta}}}\right]. & & \displaystyle\end{eqnarray}$ (8)

Therefore, the active power and reactive power of eddy current in silicon steel sheet are respectively $\begin{eqnarray} \displaystyle P_{1}\text{} & =\text{} & \displaystyle -\text{Re}\unicode[STIX]{x222F}\overset{\cdot }{\prod }\cdot ds\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle -∼\text{ah}_{s}\text{Re}\left[\overset{\cdot }{\prod }_{\left(x={\frac{d}{2}}\right)}-\overset{\cdot }{\prod }_{\left(x=-\frac{d}{2}\right)}\right]\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle \text{ah}_{s}H_{0}^{2}{\displaystyle \frac{1}{{\sigma}{\delta}}}\text{Re}\left\{(1+j)\text{th}\left[(1+j){\displaystyle \frac{d}{2{\delta}}}\right]\right\}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle \text{ah}_{s}d^{2}{\displaystyle \frac{{\omega}}{4{\delta}{\mu}}}B_{\text{av}}^{2}{\displaystyle \frac{\sinh {\displaystyle \frac{d}{{\delta}}}-\sin {\displaystyle \frac{d}{{\delta}}}}{\cosh {\displaystyle \frac{d}{{\delta}}}-\cos {\displaystyle \frac{d}{{\delta}}}}} \end{eqnarray}$ (9) $\begin{eqnarray} \displaystyle \text{Q}_{1}\text{} & =\text{} & \displaystyle \text{Im}\unicode[STIX]{x222F}\overset{\cdot }{\prod }\cdot ds\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle -\text{ah}_{s}H_{0}^{2}{\displaystyle \frac{1}{{\sigma}{\delta}}}\text{Im}\left\{(1+j)\text{th}\left[(1+j){\displaystyle \frac{d}{2{\delta}}}\right]\right\}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle \text{ah}_{s}d^{2}\frac{{\omega}}{4{\delta}{\mu}}B_{\text{av}}^{2}{\displaystyle \frac{\sinh {\displaystyle \frac{d}{{\delta}}}+\sin {\displaystyle \frac{d}{{\delta}}}}{\cosh {\displaystyle \frac{d}{{\delta}}}-\cos {\displaystyle \frac{d}{{\delta}}}}} \end{eqnarray}$ (10) Where B _av is the average magnetic induction intensity on the surface of silicon steel sheet, it can be obtained by $\begin{eqnarray}\displaystyle B_{\text{av}}=\sqrt{2}B_{0}{\displaystyle \frac{{\delta}}{d}}\sqrt{{\displaystyle \frac{\cosh {\displaystyle \frac{d}{{\delta}}}-\cos {\displaystyle \frac{d}{{\delta}}}}{\cosh {\displaystyle \frac{d}{{\delta}}}+\cos {\displaystyle \frac{d}{{\delta}}}}}}. & & \displaystyle\end{eqnarray}$ (11)

2.3. Calculation of eddy current stress voltage in silicon steel sheet

The active power P ₁ and reactive power Q ₁ of eddy current in single silicon steel sheet are as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{1}=2R_{2}I_{1}^{2}\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle Q_{1}=2{\omega}L_{2}I_{1}^{2}\end{eqnarray}$ (13) Where I ₁ is the eddy current value in silicon steel sheet.

According to the (9), (10), (12) and (13), I ₁ and L ₂ can be obtained;

The eddy current induced voltage e of a single silicon steel sheet can be approximately expressed as: $\begin{eqnarray}\displaystyle e=\sqrt{e_{1}^{2}+e_{2}^{2}}=\sqrt{(2R_{2}I_{1})^{2}+(2{\omega}L_{2}I_{1})^{2}}=\sqrt{{\displaystyle \frac{2P_{1}}{R_{2}}}\times \left({R_{2}}^{2}+\left({\displaystyle \frac{Q_{1}R_{2}}{P_{1}}}\right)^{2}\right)} & & \displaystyle\end{eqnarray}$ (14) Where, E ₁and E ₂ are the voltage drop of resistance and inductance in silicon steel equivalent circuit.

According to the above equivalent circuit, the eddy current voltage of multiple silicon steel sheets can be deduced according to the law.

3. Numerical simulation and experimental detection methods

3.1. Calculation of the eddy current loss based on the finite element method

The T, 𝜓 −𝜓 [18] equation and the 3D finite element method of equivalent conductivity and permeability tensors are used to calculate the eddy current loss of region V and verify the correctness of the calculation method of the active and reactive power, as described in Section 2.2.

As the insulation between the laminations deteriorates, the area V can be divided into the fault and non-fault areas V ₁ and V ₂, respectively. The current density J of region V can then be obtained using the potential T and magnetic potential 𝛹: $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\mathbf{J}={\nabla}\times \mathbf{T}\quad V_{1}\text{inside}\\ \mathbf{J}={\nabla}\times \mathbf{H}_{\mathbf{s}}\quad V_{2}\text{inside}\end{array} & & \displaystyle\end{eqnarray}$ (15) Where J is the current density, H _s is the magnetic field intensity produced by the external current density excitation source in infinite space. According to Boit-Savart law, H _s can be calculated as [19] $\begin{eqnarray}\displaystyle \mathbf{H}_{\mathbf{s}}={\displaystyle \frac{1}{4{\pi}}}\int _{{\Omega}_{s}}{\displaystyle \frac{\mathbf{J}_{\mathbf{s}}\times r}{r^{3}}}d{\Omega}. & & \displaystyle\end{eqnarray}$ (16)

The eddy loss can be written by $\begin{eqnarray}\displaystyle p=∼\int _{Vol}{\displaystyle \frac{J\cdot J^{\ast }}{2{\sigma}}}dVol. & & \displaystyle\end{eqnarray}$ (17)

T and 𝛹 can be solved as follows: $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\left.\begin{array}{@{}l@{}}{\nabla}\times [{\sigma}]^{-1}{\nabla}\times \mathbf{T}-{\nabla}[{\sigma}]^{-1}{\nabla}\cdot \mathbf{T}+[{\mu}]{\displaystyle \frac{\partial \mathbf{T}}{\partial t}}-[{\mu}]{\nabla}{\displaystyle \frac{\partial {\psi}}{\partial t}}=0\\ {\nabla}\cdot ([{\mu}]\mathbf{T}-[{\mu}]{\nabla}{\psi})=0\end{array}\right\}V_{1}\\ \left.\begin{array}{@{}l@{}}-{\nabla}\cdot [{\mu}]{\nabla}{\psi}=0\end{array}\right\}V_{2}\end{array} & & \displaystyle\end{eqnarray}$ (18) Where [σ] and [μ] represent the equivalent conductivity and permeability tensor of the core, respectively.

The anisotropic equivalent conductivity and permeability tensors of the non-fault area area are calculated as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle [{\sigma}]=\left[\begin{array}{@{}ccc@{}}{\sigma}_{x} & & \\ & {\sigma}_{y} & \\ & & {\sigma}_{z}\end{array}\right]=\left[\begin{array}{@{}ccc@{}}0 & & \\ & k_{Fe}{\sigma} & \\ & & k_{Fe}{\sigma}\end{array}\right]\end{eqnarray}$ (19) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle [{\mu}]=\left[\begin{array}{@{}ccc@{}}{\mu}_{x} & & \\ & {\mu}_{y} & \\ & & {\mu}_{z}\end{array}\right]=\left[\begin{array}{@{}ccc@{}}{\mu}_{fx}/(k_{Fe}+(1-k_{Fe}{\mu}_{fx}/{\mu}_{0})) & & \\ & k_{Fe}{\mu}_{fy} & \\ & & k_{Fe}{\mu}_{fz}\end{array}\right]\end{eqnarray}$ (20) Where μ_fx, μ_fy, and μ_fz represent the relative permeability of the core laminations in the x, y, and z directions, respectively, and μ₀ is the permeability of the air, and k _Fe is the superimposed pressure coefficient of the silicon steel sheets.

1. Initial fault phase: Only the insulating varnish on both sides of the initial fault area is damaged; hence, the conductivity on both sides of the fault area is set as σ, and the internal conductivity is still a nonfaulty conductivity.

2. Complete fault phase: According to the basic principle of the homogenization method, the anisotropic equivalent conductivity and permeability tensors of the complete fault area are calculated as in [20,21]. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle [{\sigma}]=\left[\begin{array}{@{}ccc@{}}{\sigma}_{x} & & \\ & {\sigma}_{y} & \\ & & {\sigma}_{z}\end{array}\right]=\left[\begin{array}{@{}ccc@{}}{\displaystyle \frac{l^{2}\times {\pi}\times f\times {\mu}_{r}}{k\times {R_{q}}^{2}\times {b_{t}}^{2}}} & & \\ & k_{Fe}{\sigma} & \\ & & k_{Fe}{\sigma}\end{array}\right]\end{eqnarray}$ (21) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle [{\mu}]=\left[\begin{array}{@{}ccc@{}}{\mu}_{x} & & \\ & {\mu}_{y} & \\ & & {\mu}_{z}\end{array}\right]=\left[\begin{array}{@{}ccc@{}}{\mu}_{fx} & & \\ & k_{Fe}{\mu}_{fy} & \\ & & k_{Fe}{\mu}_{fz}\end{array}\right].\end{eqnarray}$ (22)

Where R _q is core lamination resistance, k is the temperature coefficient of insulating material, f is the frequency of fault current. And R _q = 2nR ₁ + 2R ₂ n is the number of faulty silicon steel sheets, which R ₁, R ₂ can be calculated from formula (1).

3.2. Induced voltage acquisition experiment based on the eddy current detection

An experimental detection method is used for comparative analysis, and the same fault area as the analytic method is selected as the fault area detection point of the insulation area to verify the correctness of the calculation method for solving the induced voltage, as described in Section 2.3.

Simulation of the Fault Area:

1. Initial fault phase: Destroy the insulating varnish on the two sides of the stator tooth (Fig. 2(b)) into three, four, five, and six pieces with an electric drill and use the material with the same conductivity and permeability as the silicon lamination to weld the two sides of the damaged pieces and ensure the formation of the fault current circuit.

2. Complete fault phase: Completely destroy the insulating varnish on the top of the stator tooth into three, four, five, and six pieces with an electric drill (damage length: Lw) and use the material with the same conductivity and permeability as the silicon lamination to weld the damaged area and ensure the formation of the fault current circuit.

When the stator core of a large motor is excited, an alternating magnetic field will be generated in the silicon steel sheets of the iron core. If the insulation between the laminations is damaged, the damaged site and the pigeon tail positioning tendon on the back of the core will form a closed loop. Further, the magnetic field at the silicon steel sheet will be affected by the resulting short circuit current, which will lead to the distortion of the magnetic field lines passing across the damaged area.

According to Faraday’s law of electromagnetic induction, the characteristic parameters of the equivalent current source can be determined as the change in magnetic flux through the coil is known, that is, deterioration parameter that is, induced voltage in inter chip fault region. The experimental principle is shown in Fig. 7.

Fig. 7.

Experimental schematic diagram.

The detection coil is made of permalloy material with high initial permeability and maximum permeability, meanwhile, in order to reduce the eddy current loss in the permalloy and increase the measurement accuracy of the detection coil, a Φ0.1 mm permalloy wire was bundled in the same thickness, and each permalloy wire was coated with an insulating paint when the second coil was manufactured.

Considering that the detection of the second coil is carried out in the movement, the second coil was placed on the moving trolley with the aluminum alloy as the frame, and the screw-clamping base, the roller, and the permanent magnet were fixed on the base of the car for fixing the second coil. Further, a position encoder and its control box were installed on the side of the front and back of the car, respectively, which can ensure that the trolley could move smoothly along the axial direction of the tooth. The specific experimental platform is show in Fig. 8.

Fig. 8.

Experimental platform.

The white coil in Fig. 8 is the excitation coil located at the axis of the stator cavity. The distance between the end of the stator core and the excitation coil is greater than 1 m. The regulator of the excitation system regulates the voltage required by the excitation system to produce a magnetic field in the stator core. The trolley carrying a sensor probe scans the inner core surface, collects the information of the magnetic field of the stator core in real time, detects the change in the magnetic field at the fault between the laminations, displays the voltage waveform on the upper computer through the processing component of the detection system, and separately extracts the voltage data under different fault conditions.

4. Results of the calculation and contrastive analysis

4.1. Experimental prototype parameters

This study considered a large high-voltage motor as the prototype whose insulation grade is F. The cooling mode is IC01. The iron core silicon steel sheet model is DR530-50. The silicon steel sheet’s lamination coefficient is 0.95. The conductivity σ under rated conditions is 5 MS/m, and the relative permeability μ_r is 2000. The single silicon steel sheet’s thickness is 0.5 mm.

The insulation between the core laminations of the prototype is polyamine imide (enamel) whose heat resistance grade is 155 °C (F). Figure 9 shows the two-dimensional curve of the volume resistivity of the insulating materials 𝜌_v changing with the environment temperature [22].

Fig. 9.

Change curve of volume resistivity.

4.2. Comparison between the analytical and numerical simulation results

The eddy current loss per volume was calculated based on the equivalent circuit of the failure domain in Figs 5 and 6. A field calculator was then used to calculate the eddy current loss per volume of the initial and complete failure domains. Figure 10 illustrates the comparison between the analytical calculation and the simulation results.

Fig. 10.

Comparison between the results of analytical method and the results of finite element simulation.

The deviation between the analytical calculation and the finite element simulation results was less than 5%. The eddy current loss in the complete failure domain was larger than that in the initial failure domain when the number of the faulted laminations was the same. Moreover, the eddy current loss gradually increased as the number of faulted laminations increased.

4.3. Comparison between the analytical and experimental results

The induced voltage in the failure domain was calculated based on the equivalent circuit of the failure domain in Figs 5 and 6. An experimental test on the induced voltage in the failure domain was conducted. Figure 11 shows the comparison between the analytical calculation and experimental experimental results.

Fig. 11.

The comparison between the calculation results of analytical method and the experimental results.

The analytical calculation results of the initial and complete failure domains had the same trend with the experimental ones: the induced voltage in the complete failure domain was larger than that in the initial failure domain when the number of faulted laminations remained the same, and the induced voltage in the failure domain increased as the number of faulted laminations increased. The analytical calculation results were always smaller than the experimental ones. One of the reasons for such a phenomenon is the following: the axial resistance R ₁ of the single silicon steel sheet in the equivalent circuit of the failure domain was not considered during the analytical calculation. Moreover, the assumed conditions in the analytical calculation and some factors during the experiment process caused a relatively big deviation between the experimental and analytical calculation results. However, the two results had the same trend; hence, this deviation would not hinder the application of the algorithm in engineering practices.

5. Usage of grey theory to predict the axial damage depth of the iron core

Considering the difference in the orders of magnitude between the mechanical structure of the iron core and the size of the insulation thickness in the failure domain, the eddy current–voltage variation of the combinatorial model of the silicon steel sheet and the insulation between the core laminations was used to simulate the variation of the effective insulation thickness Δ and predict the axial damage depth of the iron core.

5.1. Building the grey model GM(1,1)

The stator core is an integral component of a large high-voltage motor. The strength of its insulation between laminations will gradually decrease with the change in the operating environment and time. Meanwhile, very few fault data sample and no typical distribution laws exist because of the dismantling operation and certain destructiveness and difficulties in the experimental test. However, the grey theory prediction method can provide approaches for solving systematical problems in the absence of information and for predicting the evolution law of the axial damage depth of the iron core.

The analytical calculation results of the induced voltage in Section I-A-3 were used to construct the induced voltage sequence varying with the number of faulted laminations. $\begin{eqnarray}\displaystyle E^{(0)}=\{E^{(0)}(3),E^{(0)}(4),\cdots ∼,E^{(0)}(n)\}. & & \displaystyle\end{eqnarray}$ (23)

An accumulation of E ⁽⁰⁾ to generate a series of approximate exponential law changes are made, $E^{(1)}=\{E^{(1)}(3),E^{(1)}(4),\cdots ∼,E^{(1)}(n)\}$ , where n is the number of fault laminations, n = 3, 4, ⋯ .

The corresponding whitening differential equation of E ⁽¹⁾ is established as follows: $\begin{eqnarray}\displaystyle {\displaystyle \frac{dE^{(1)}}{dt}}+{\alpha}E^{(1)}={\mu} & & \displaystyle\end{eqnarray}$ (24) Where, 𝛼 is the development coefficient; μ is the grey action quantity.

The least squares of the [𝛼, μ]^T is done by $\begin{eqnarray}\displaystyle [{\alpha},{\mu}]^{\text{T }}=(B^{\text{T }}B)^{-1}B^{\text{T }}Y. & & \displaystyle\end{eqnarray}$ (25) In order to do an accumulation to generate a sequence of approximate exponential rule changes, where is the number of fault laminations.

Perform the least squares calculation on: $\begin{eqnarray}\displaystyle Y=\left[\begin{array}{@{}c@{}}E^{(0)}(4)\\ E^{(0)}(5)\\ \vdots \\ E^{(0)}(n)\end{array}\right],\quad B=\left[\begin{array}{@{}cc@{}}-{\displaystyle \frac{1}{2}}(E^{(1)}(3)+E^{(1)}(4)) & 1\\ -{\displaystyle \frac{1}{2}}(E^{(1)}(4)+E^{(1)}(5)) & 1\\ \vdots & \vdots \\ -{\displaystyle \frac{1}{2}}(E^{(1)}(n-1)+E^{(1)}(n)) & 1\end{array}\right]. & & \displaystyle\end{eqnarray}$ (26)

The prediction equation can be obtained by solving $\begin{eqnarray}\displaystyle \overset{\wedge }{E}^{(1)}(i+1)=\left(E^{(0)}(1)-{\displaystyle \frac{{\mu}}{{\alpha}}}\right)e^{-{\alpha}i}+{\displaystyle \frac{{\mu}}{{\alpha}}} & & \displaystyle\end{eqnarray}$ (27) Where i = 3, 4, ⋯ , n −1.

5.2. Predicting the damage depth and comparing the experimental results

Through software calculation, the prediction equation under initial failure can be obtained by: $\begin{eqnarray}\displaystyle \overset{\wedge }{E}_{\text{the initial fault phase}^{(i+1)}}^{(1)}=138.792507e^{0.1516i}-140.4221. & & \displaystyle\end{eqnarray}$ (28)

The prediction equation under complete failure is as follows: $\begin{eqnarray}\displaystyle \overset{\wedge }{E}_{\text{the complete fault phase}^{(i+1)}}^{(1)}=158.6198589e^{0.1423i}-160.1264. & & \displaystyle\end{eqnarray}$ (29)

A b–b reduction to formulas (28) and (29) was made to obtain two failure modes. Figure 12 shows the development curve of the faulted core laminations and its comparison with the experimental test data when the number of faulted laminations was from 7 to 12.

Fig. 12.

The comparison between the calculation results of analytical method and the experimental results.

Figure 12 shows that the prediction results from the analytical method had the same trend as those from the experiment, and the numerical deviation of the two results was within 7%. The induced voltage in the failure domain nonlinearly increased as the number of faulted laminations increased.

6. Conclusions

An analytical method was used to quantitatively calculate the failure state and the electric parameters of the failure domain between the iron cores of the large high-voltage motor. The results were verified through a numerical simulation and an experimental test to show that the calculation results meet the actual engineering demands. The analytical method is not only a simple and fast computing method but its solving process can also intuitively reflect the eddy current loss and the induced voltage variation between the faulted laminations of the silicon steel sheet, thereby providing a data reference for the optimized design of local structures. Meanwhile, a grey theory method was used to predict the axial damage depth between the faulted laminations: the induced voltage in the failure domain surged and the axial damage depth nonlinearly increased as the number of faulted laminations increased.

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