Parametric Study and Improvement of Anti-Corona Structure in Stator Bar End Based on Finite Element Analysis

Abstract

Voltage withstand tests on stator bars can cause destructive phenomena such as thermal breakdown and flashover discharge on the surface of the anti-corona layer. This study optimizes the anti-corona structure at a stator bar’s end to prevent such failures using a 120 MW water-cooled turbogenerator with a rated voltage of 15.75 kV. For a well-designed anti-corona system, the maximum potential gradient of the stator bar should be lower than the discharge intensity of air corona. In our design, the electric field intensity is maintained below 3.1 kV/cm, and the maximum surface loss in the anti-corona layer is limited to less than 0.6 W/cm². Additionally, the terminal voltage is kept lower than that of flashover voltage at rated conditions. Furthermore, the length of the anti-corona layer should be minimized. The optimization process involves determining the rotation angle of the stator bar, calculating the total length of the anti-corona layer, and analyzing the electric field and loss in the layer at different lengths. The results demonstrate that the optimized anti-corona design effectively reduces the risk of flashover and thermal failure, ensuring stable operation under rated conditions. This manuscript belongs to purely computational experiments. At present, the electrical machinery with 120 MW rated power grade is put into operation steadily. There is a growing requirement for anti-corona. In this manuscript, computing method is used to assist the anti-corona structure design. The electrical machinery insulation is improved by better anti-corona materials. Therefore, the service life of electrical machinery can be prolonged, which is significant in engineering.

1. Introduction

The loss and heat generated by motor insulation directly determine the service life of high-voltage motors. The stator winding is a core component of large generators and, during operation, must withstand the effects of heat, electricity, and environmental conditions [1,2]. Therefore, the design of insulation systems in stator windings plays a critical role in determining the overall insulation level of electrical machinery. To ensure sufficient mechanical strength for motor insulation, the thickness of the main insulation must be increased [3,4]. However, to improve heat dissipation and reduce the size of the alternator, the main insulation thickness should be minimized, creating a design conflict [5]. To obtain the optimum main insulation thickness, finding the suitable anti-corona structure and methods is essential. In high-voltage electrical machinery, corona discharge mainly occurs in the air gap between the main insulation with a groove or the end of the stator winding. When the voltage is sufficiently high, even though the average field strength is low, the electric field intensity in these parts reaches a high value [6,7]. In such localized regions, air discharge occurs, leading to corona formation [8]. During this process, the stator winding current and corona current interact, generating heat, sound, and light [9]. The corona power loss results in overheating of electrical machinery [10]. Additionally, chemical reactions occur, producing ozone and nitrogen oxides, which corrode the insulation system and severely damage its properties. Finally, this leads to the breakdown of the main insulation, reducing the reliability of insulation of the electrical machinery and significantly shortening its service life.

The main insulation material directly affects the service life of electrical machines. The main insulation typically consists of mica, an adhesive, and a reinforcing material. In the 1960s, Yuan introduced the bitumen mica tape dipping insulation system, which became widely used in motor insulation by the 1980s. By the 1990s, the thermosetting epoxy and polyester resin mica tapes were used as the main insulation material, leading to the development of EP mica tape, which gained approval from major motor manufacturers by 2000 [11,12]. Building on this, continuous improvements were made with epoxy mica insulation. In this new insulation system, thermosetting adhesive replaced bituminous paint and glue, alkali-free glass replaced paper as the reinforcing material, and mica powder replaced mica splitting [13]. Additionally, advancements in insulation structure technology further enhanced its performance. This new insulation material offered great advantages regarding durability and efficiency [14].

The stator bar of a high-voltage motor is located outside the groove, further from the core, which results in a higher ground potential for the winding [15]. Underrated voltage and current can flow along the surface of the conductor from the wire bar to the iron core, leading to potential discharge at the wire bar end due to the concentration of the electric field [16]. To mitigate this issue and reduce discharge, anti-corona treatment is applied to the wire bar end, ensuring that the electric field is evenly distributed [17]. This prevents phenomena such as discharge, corona, and overheating during the normal operation of an alternator [18]. With the increasing deployment of electrical machinery with 26–27 kV, the requirements for anti-corona measures have become more stringent. By applying rational computational methods, it is possible to design optimal anti-corona structures and select improved anti-corona materials. Enhancing insulation and prolonging the service life of electrical machinery remain critical focus areas. Extensive computational methods based on electric circuits, electromagnetic fields, and finite element theory have been explored to analyze the electric field at the ends of high-voltage machinery and optimize anti-corona structures. These methods have produced favorable results in many applications [19]. In 1967, A. Kelen proposed an empirical formula for nonlinear anti-corona coatings. In 1984, Yoshifumi introduced a time-periodic method to analyze silicon carbide (SiC) anti-corona coatings, accounting for the nonlinearity of silicon carbide and the time factor. In 1986, Yun used the Newton–Raphson method to optimize corona protection structures, offering easy model establishment and fast computational speed; however, it did not account for the spatial distribution of the electric field, leading to some errors due to the assumption of a pure capacitive relationship between the conductor and semiconducting anti-corona coating [20,21]. In recent years, the rapid development of computer technology has led to the widespread use of finite element simulation in optimizing the stator bars of electrical machinery. In 2018, Yuan explored numerical simulations of wire bars, modeling their position within the interphase electric field, but the accuracy of these calculations remains limited [22]. In 2022, Yuan applied the finite element method to analyze rotating machines, using a transient solving approach to obtain the nonlinear electric field distribution. However, the accuracy and efficiency of the method remained suboptimal. Earlier, in 2014, Yuan developed an electric field distribution program for a 27 kV electrical machinery stator bar using MATLAB 7.0. The electric field distribution and associated losses were further analyzed using COMSOL 6.1 software, yielding ideal results [23,24].

During the voltage withstand test, thermal breakdown and flashover discharge will happen in the anti-corona layer surface of stator bar [25]. The purpose of anti-corona structure optimization in stator bar end is preventing these destructive phenomena. The excellent anti-corona design need to meet the following conditions [26].

(1) The maximum potential gradient must be lower than the strength of air corona discharge. The air maximum flashover field strength is 8.1 kV/cm. In practical application, the electric field intensity is lower than 3.1 kV/cm.

(2) The maximum surface loss value in anti-corona coatingis lower than 0.6 W/cm².

(3) Terminal voltage on conductor potential is lower than flashover voltage value under rated voltage.

(4) Under possible operating conditions, the length of anti-corona layer should be as short as possible.

Under these conditions, the anti-corona structure of a hydro generator with 120 MW rated power and 15.75 kV rated voltage can be designed and optimized. Firstly, the rotation angle of wire bar should be defined, from which the total length of anti-corona layer is obtained. After the calculation of electric field and loss in the anti-corona layer with different length, the intrinsic resistivity and nonlinearity coefficient of anti-corona material are adjusted, from which the stator bar can be optimized. This manuscript discusses purely computational experiments. At present, the electrical machinery with 120 MW rated power grade is put into operation steadily. There is a growing requirement for anti-corona. In this manuscript, the computing method is used to assist the anti-corona structure design. The electrical machinery insulation is improved by better anti-corona materials. Therefore, the service life of electrical machinery can be prolonged, which is significant in engineering.

2. Materials and Methods

2.1. Anti-Corona Technique of Stator Bar

The end structure of large scale hydro generator mainly consists of a stator bar, stator core, the main insulation, and anti-corona coating [27]. It is a kind of insulation structure with a sleeve type. The electric field distribution in this sleeve type structure is uneven [28]. When the value of electric field is high enough, corona discharge will happen. The prolonged corona discharge will damage the insulation structure, which causes insulation breakdown [29]. The electric field formed by this structure is extremely uneven. Therefore, the reliability of motor operating is a serious influence [30].

The electric field uniform distribution in the exit slot of the large generator reduces the chance that corona discharge happens. The coil end is wrapped by one or several layers of a silicon carbide (SiC) anti-corona belt, from which the electric field distribution can be improved. SiC possesses a nonlinear characteristic. With the electric field increasing, the resistivity decreases. With this feature, the electric field in the exit slot can be automatically adjustment to homogenization, from which the corona discharge incidence decreases. The nonlinear resistance characteristics of SiC can be expressed in Equations (1) and (2).

$$\rho ={\rho }_0\exp (-\beta E)$$

(1)

$$\rho ={\rho }_0\exp (-\beta \left|{\displaystyle \frac{d_u}{d_x}}\right|)$$

(2)

Among them,

${\rho }_0$: When E = 0, the intrinsic resistivity of SiC.

$\rho$: The resistivity of SiC corresponding to field strength E.

$\beta$: Nonlinearity coefficient of SiC, of which unit is m/V.

E: Electric field intensity of SiC suffered.

After the anti-corona coating is painted onto the surface of the main insulation, according to different computing methods, the electric field distribution and anti-corona optimization at the end of the large generator is obtained. It provides a certain reference for anti-corona structure design of the electrical machinery. From this basic structure, lots of new computing methods based on electric circuit and electromagnetic theory are derived. The calculation result is accurate.

The optimization and design of the electric field at the ends of large electrical machinery have long been a focus of interest among scholars. Numerous theories and calculation methods have been proposed based on electric field analysis and calculation. In recent years, more practical methods have emerged for calculating higher voltage levels in electrical machinery.

The structure of the stator bar end is shown in Figure 1, and it can be simplified into the equivalent circuit shown in Figure 2. In this circuit, C_v is the unit length capacitance of the wire bar main insulation. $\rho$ is the unit length surface resistivity of the wire bar. The current flows from the off-side slot outlet to the wire bar groove. As shown in Figure 2, the current flowing through the resistance in the off-side slot outlet is minimal. However, as the distance from the notch increases, more current accumulates in the wire bar. If ${\rho }_1={\rho }_2={\rho }_3=\cdots \cdots$, and $I_1=I_2=I_3=\cdots \cdots$, the voltage drop, $∆u_1$, across ${\rho }_1$ exceeds that at the off-side slot outlet. Consequently, the voltage at the slot outlet increases, leading to a stronger electric field at that location.

These characteristics can be expressed mathematically. Let x denote the distance from any point on the wire bar to the notch. The potential difference at each point on the surface relative to the copper busbar is $U_x$, and the surface current is $I_x$. According to Figure 2, Equation (3) can be expressed as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{{d^{2}U}_x}{d{x}^2}}-\alpha U_x=0\\ x=0,U_x=0\\ x=l,{\displaystyle \frac{du}{dx}}=0\end{array}\right.$$

(3)

Equation (3) can be converted to Equation (4).

$$\left\{\begin{array}{c}{U}_x={\displaystyle \frac{cosh\alpha (L-x)}{cosh\alpha L}}\\ I_x={\displaystyle \frac{u}{\rho }}\alpha th\alpha (L-x)\end{array}\right.$$

(4)

In Equation (4), $\mathsf{\alpha }=\sqrt{\omega C_V\rho }$.

From Equation (4), as the notch distance, x, increases, both $U_x$ and $I_x$ decrease. The electric field becomes more concentrated at the notch.

2.2. Corona Protection Material of Stator Bar

The main insulation plays a crucial role in determining the performance parameters and ensuring the stable operation of electrical machinery. When the main insulation of the stator winding is destroyed, it can lead to short-circuit faults at the ends of the machinery. This damage may cause copper conductors to liquefy and insulation materials to carbonize, ultimately producing conductive substances that can result in widespread short circuits. Therefore, this manuscript explores the factors contributing to insulation faults. Reducing dielectric breakdown is crucial for decreasing electrical machinery faults, which is of great significance for power production safety. The insulation properties of electrical machinery can be affected by factors such as insulation binding, the curing process, and the heat effects on winding insulation. To improve the insulation properties, three aspects are discussed below. First, during the insulation binding process, the traverse angle radius should be larger than 2 mm to ensure proper insulation. Second, the technology parameters used in the curing and molding process must be adjusted according to environmental conditions. In general, a dry mica tape vacuum varnish insulation process is employed. The thermal resistance of the insulation must be designed to manage heat radiation effectively. Additionally, the temperature surrounding the electric generator must be closely controlled.

To reduce the occurrence of corona discharge and achieve an even distribution of the electric field at the ends of electrical machinery, the development of a new type of corona protection material is essential. The material’s resistance should decrease as the electric field increases, allowing for automatic adjustment of the electric field. In the 1990s, anti-corona varnish was developed by mixing carbon black, black lead, and pitch. This varnish was applied to the surface of a wire bar groove, reducing the potential gradient at the coil ends and eliminating corona discharge. However, the breakdown field strength of this anti-corona coating was approximately 2 kV/cm, making it prone to surface breakdown. As voltage levels increased, this varnish could no longer meet the requirements for higher voltages. Subsequently, semiconductive SiC was identified as a superior material. A semiconductive varnish composed of SiC and organic varnish has since been used as a corona protection material for electrical machinery. This SiC-based material is still in use today. SiC is a crystalline material with a high breakdown field strength, stable chemical properties, a high thermal conductivity coefficient, and a high electron saturation velocity. The resistivity of this corona protection material has an exponential relationship with the electric field intensity, a nonlinearity that is exploited in corona protection design. By selecting appropriate values for the anti-corona coating, the electric field can be effectively homogenized.

Two main factors affect the inception voltage of the anti-corona coating: surface resistivity and the nonlinearity coefficient. These properties are mainly affected by the size and content of the SiC particles, as well as the impurities and SiC content. Experimental results show that for α-SiC, as the particle size increases, resistivity decreases, and the nonlinearity coefficient increases. Therefore, SiC with larger particle sizes can be used as a medium-resistance corona protection material, while SiC with smaller particle sizes is suitable for high-resistance anti-corona varnishes. Additionally, higher SiC content results in lower resistivity and higher nonlinearity coefficients. The presence of trivalent metal impurities decreases both the resistivity and nonlinearity coefficient of the anti-corona varnish, while pentavalent metal impurities increase these properties.

The electrical properties of different types of SiC vary significantly. Experimental results show that adding β-SiC to α-SiC reduces the resistivity and increases the nonlinearity coefficient. When the content of β-SiC is 7.5%, the resistivity decreases from 7.89 × 10⁷ Ω·m to 1.34 × 10⁷ Ω·m, while the nonlinearity coefficient rises from 1.527 to 1.695. If this anti-corona system is applied to the stator bar of electrical machinery, the inception voltage increases by 35%.

2.3. Primary Methods for Adjusting the Electric Field at the Ends of Electrical Machinery

Based on the above analysis, the notch at the end of electrical machinery is the primary location of electric field concentration and corona generation. Homogenizing the electric field at the ends of electrical machinery can effectively suppress corona generation. There are two primary methods for adjusting the electric field at the ends of electrical machinery:

(1) Electric field adjustment by resistance divider

Under alternating voltage, the electric field distribution in the stator bar is not solely determined by leakage resistance. The capacitive current between the stator core and coil is much larger than the conductive current, and the main insulation resistance is typically assumed to be infinite. By applying a layer of semiconductive varnish to the main insulation, the electrical conductivity can be increased, thereby lowering the surface resistivity of the insulation layer. This causes a voltage drop around the notch, thereby modifying the end electric field. In high-voltage electrical machinery systems, one or multiple sections of semiconducting anti-corona varnish are brushed onto the stator bar, to improve the end electrical field. Most electrical machinery manufacturers use resistive voltage-sharing techniques to manage anti-corona treatment at the stator winding end.

(2) Electric field adjustment by capacitor equal-press method (inner shield method)

Under normal conditions, a conductor in a uniform electric field acts as an equipotential surface, where the voltage remains constant throughout. However, if an electrode is introduced into the electric field, the original uniform electric field is no longer the equipotential surface. The distribution of the electric field will change to a certain extent. By this the electric field can be adjusted. A semiconducting layer is placed into the insulation layer of the electrical machinery notch, from which two capacitors form. This is the capacitor equal-press method. From this method, a telescopic electric field structure is established, making the electric field more uniform, and it is also known as the inner shield structure. In this approach, a metal layer is embedded within the insulation layer. However, the process is complex, and during the shaping of the wire bar, shrinkage stress can damage the shielding layer, potentially leading to short circuits and damaging the main insulation. For this reason, the inner shield method is less commonly used.

3. Results

3.1. Wire Bar Corner

Because COMSOL Multiphysics6.0 has great functions, in this manuscript, the AC/DC module is invoked. This module includes a steady-state and dynamic state electromagnetic field in a two-dimensional and three-dimensional space. In addition, the passive and active components based on circuit modeling are also included. All modeling formulas can be built based on Maxwell equations, subsets, exceptions, and Ohm’s law [29]. According to a series of predefined physical field interface (the special physical field interface for low frequency electromagnetics), the application model of electric field, magnetic field and electromagnetic field under steady state and low frequency can be built. The modeling process follows these steps: geometry define, materials selection, low frequency electromagnetic field and physical field interface selection, boundary and initial conditions definition, finite element network definition, and solver and visualization results selection. All these steps can be carried out in COMSOL [30]. With default settings in solver, these steps are auto completion. In addition, COMSOL Multiphysics6.0 possesses an abundant post processing function. Various data, figures, and curves can be analyzed based on users’ requirement.

Wire bar bending leads to an inhomogeneous distribution of the electric field, electric potential, and loss, especially affecting surface loss. The heat generated by the narrow face and wide face losses in a wire bar differs. Experimental results show that adjusting the flat turn angle can improve the electric field and loss distribution to some extent, but it is ineffective for optimizing the entire anti-corona system. In addition, increasing the motor size negatively impacts its structure.

Another important factor is the arc corner angle of the wire bar, which directly affects the electric field and loss distribution. Additionally, the space occupied by the wire bar is reduced. Therefore, the effect of the arc corner angle on the electric field distribution of the anti-corona coating requires further discussion.

Three wire bars with different arc corner angles are compared and analyzed in terms of electric field, electric potential, and loss. The corona protection structure and material properties of these test wire bars are identical, while the arc corner angles are set at 15°, 17.5°, 20°, 22.5°, 25°, 27.5°, and 30°. The three-dimensional electric field distribution diagrams for wire bars with arc corner angles of 15° and 22.5° are shown in Figure 3.

From the three-dimensional electric field distribution diagram, the color of the anti-corona coating at the beginning of the medium-resistance region is red, gradually transitioning to blue. The color change in the first section of the anti-corona coating is less pronounced, which is attributed to differences in the nonlinearity coefficient. At the junction between the medium resistance and medium-high resistance regions, the color shifts from red to blue again. While the 3D electric field distribution is similar across all angles, the highest electric field values differ. As shown in Figure 4, the line chart illustrates the highest electric field intensity corresponding to each arc corner angle. The maximum electric field intensity for all seven arc corner angles is below 3.0 kV/cm, which meets the anti-corona requirements. When the arc corner angle is 15°, the highest electric field value is 2.89 kV/cm. At an angle of 22.5°, the highest electric field value drops to 2.49 kV/cm, a reduction of 13.8%. This demonstrates that the anti-corona effect is satisfactory.

The loss distribution for wire bars with arc corner angles of 15° and 22.5° is shown in Figure 5.

According to the figure, the maximum loss value is mainly focused at the beginning of the first section of the anti-corona coating. There are noticeable differences between the inner and outer corners. For the wire bar with a 15° arc corner, the highest loss value reaches 0.72 W/cm². The resistance losses for the wire bars across all angles are similar, with a clear concentration of loss in the wide inner corner. Upon comparison, the highest loss value for the stator bar with the 22.5° arc corner is 0.47 W/cm², which is 34.7% lower than the loss value for the stator bar with the 15° arc corner. The distribution curve illustrating the highest surface loss across all arc corner angles is shown in Figure 6.

According to the comparison, when the arc corner angle is set to 22.5°, both the highest electric field intensity and maximum loss value are lower, and the electric field distribution is more uniform. By contrast, for an arc corner angle of 15°, the highest electric field and loss values are 9% and 34.7% higher, respectively. Therefore, the anti-corona effect is optimal when the arc corner angle is 22.5°.

3.2. Length of Anti-Corona Coating

To reduce costs, the length of the anti-corona coating should be kept as short as possible. However, the terminal voltage between the anti-corona coating and the conductor potential must remain lower than the flashover voltage underrated conditions. To prevent flashover, it is important to examine how changes in the length of the anti-corona coating affect the electric field in different resistive layers. From the 3D electric field distribution curve, the areas of concentrated electric field occur at the exit slot and at the junction between the medium resistance and medium-high resistance layers. Therefore, a detailed analysis of how changes in the anti-corona coating length impact the electric field is necessary.

To intuitively observe the effect of anti-corona coating length on the electric field, a one-dimensional analysis is applied to the electric field distribution. Since the anti-corona coating forms a curved surface in space, there are differences in length on each side, making direct observation of the coating’s changes challenging. In practical applications, the length of the anti-corona coating is adjusted by controlling the cutting surface group distance, with 20 mm set as one cutting interval. For testing purposes, a continuous line is used on the surface of the anti-corona coating, as shown in Figure 7.

The variations in the anti-corona coatings for medium resistance and medium-high resistance are shown in

Table 1. Length of anti-corona coating for change medium resistance and medium-high resistance.

	Anticorona Coating	Medium Resistance Section (mm)	Medium-High Resistance Section (mm)	High Resistance Section (mm)	Color
Groups
Original length		170.48	183.13	232.58	Black
1		192.03	161.58	232.58	Pink
2		213.57	140.04	232.58	Blue
3		235.18	118.49	232.58	Green
4		256.66	96.95	232.58	Red
5		278.21	75.40	232.58	Cyan

.

The variation in the electric field with the lengthening of the medium resistance and shortening of the medium-high resistance is shown in Figure 8. The explanation of the colors is in Table 1. As the medium resistance lengthens and medium-high resistance shortens, the electric field intensity at the junction between the two layers decreases. Before these changes, the electric field at the junction between the medium resistance and medium-high resistance reached 3.46 kV/cm. After adjusting the medium-resistance length, the highest electric field intensity at the interface decreased to 3 kV/cm. Additionally, changes in the medium-high resistance affected the highest electric field intensity in the anti-corona coating.

The highest electric field values corresponding to each length variation are summarized in

Table 2. Maximum electric field after changing resistance length.

Groups	Original Length	1	2	3	4	5
Highest electric field values (kV/cm)	4.51	4.65	4.05	4.70	4.76	4.22

.

From Table 2, the highest electric field in group 2 decreases the most, reaching 4.05 kV/cm, compared to the original length of the medium resistance, where the highest electric field was 4.51 kV/cm. This indicates that the anti-corona length in group 2 is the most satisfactory.

To further optimize the electric field distribution, the length of the medium resistance is kept unchanged, while the lengths of the medium-high resistance and high resistance are adjusted. The resulting electric field changes are analyzed using the same method. The variations in the anti-corona coatings for medium-high resistance and high resistance are shown in

Table 3. Length of anti-corona coating for change medium-high resistance and high resistance.

	Anti-Corona	Medium Resistance Section (mm)	Medium-High Resistance Section (mm)	High Resistance Section (mm)	Color
Coating Groups
Original length		170.48	183.13	232.58	Black
6		170.48	204.67	211.04	Pink
7		170.48	226.22	189.50	Green
8		170.48	247.76	167.75	Red
9		170.48	269.30	146.40	Cyan

.

The distribution of electric field variations corresponding to each anti-corona coating length after adjusting the medium-high and high resistance lengths is shown in Figure 9. The explanation of the colors is in Table 3.

The highest electric field value of each anti-corona length is shown in

Table 4. Maximum electric field after changing medium-high resistance length.

Groups	Original Length	6	7	8	9
Highest electric field values (kV/cm)	4.49	4.28	5.33	4.74	4.40

.

From Table 4, the highest electric field value in group 6 decreases significantly, being 4% lower than that in the original length group. However, in group 7, the changes in the anti-corona length result in an increase in the maximum electric field value to 5.33 kV/cm, which is 15% higher than that in the original length group. This increase is unfavorable for electric field homogenization.

Subsequently, the changes in electric field and potential after shortening the high resistance length were explored, yielding similar conclusions. Based on the analysis of anti-corona length, the optimal anti-corona coating length for the large generator stator bar is shown in

Table 5. Optimal length distribution.

Parameters	Medium Resistance Section	Medium-High Resistance Section	High Resistance Section
Length (mm)	213.57	204.67	167.95

, where the electric field distribution homogenization is most effective.

4. Discussion

4.1. Electrical Resistivity Setting

Semiconducting SiC with nonlinear properties is selected as the anti-corona coating material. According to the potential distribution, the surface potential gradually increases from the medium-resistance region to the anti-corona end, followed by a sharp increase before eventually decreasing. This pattern results from the decreasing surface current. To maintain uniformity in the electric field, the resistance value of the anti-corona coating increases gradually. Therefore, the resistance values, from lowest to highest, are medium resistance, medium-high resistance, and high resistance. The surface resistivity of semiconductive materials furthest from the exit slot is the highest. A detailed discussion on selecting the appropriate resistivity of semiconductive materials and maximizing their properties is provided.

Regarding electric field distribution, a sudden change in the electric field is likely to occur at the junction between the starting point of the anti-corona coating, medium resistance, and medium-high resistance. These areas experience the most serious heating and loss phenomena. Because the current at the end of the high resistance is low, its effect on loss distribution is less significant. The effect of changing the medium resistance and medium-high resistance on the electric field and loss distribution is explored. The electrical resistivity of low resistance is fixed at 0.02 Ω·m, while the resistivity values of medium resistance, medium-high resistance, and high resistance range from 10⁶ to 10¹² Ω·m. After a comprehensive analysis of the nonlinearity coefficient and the influence of an electric field on resistivity, it was found that the resistivity of the anti-corona coating can adjust by a factor of up to 10² under the effect of the electric field. The resistivity values for each anti-corona coating layer are shown in

Table 6. Surface resistivity of anti-corona coating.

	Anti-Corona Coating	Medium Resistance Section (Ω·m)	Medium-High Resistance Section (Ω·m)	High Resistance Section (Ω·m)
Groups
1		8 × 106	5 × 107	1 × 108
2		8 × 106	5 × 108	1 × 1010
3		8 × 106	5 × 109	1 × 1012
4		8 × 107	5 × 108	1 × 109
5		8 × 107	5 × 109	1 × 1011
6		8 × 107	5 × 1010	1 × 1013
7		8 × 108	5 × 109	1 × 1010
8		8 × 108	5 × 1010	1 × 1012
9		8 × 108	5 × 1011	1 × 1014

.

The values between each group are determined through simulation calculations, and the maximum loss values are shown in

Table 7. Maximum value of surface loss after changing resistivity.

Groups	1	2	3	4	5	6	7	8	9
Maximum value of surface loss (W/cm2)	0.511	0.578	0.635	0.430	0.536	0.752	0.492	0.699	0.792

.

The comparison of surface loss distribution in group 4 and 9 is shown in Figure 10.

Figure 11 depicts the changes in the highest surface loss values for different electrical resistivities. Based on the test results and analyses, it is observed that when there is one-order-of-magnitude difference in resistivity between adjacent layers, the surface loss is minimized. Conversely, when the difference in resistivity between layers is three orders of magnitude, the highest surface loss exceeds 0.6 W/cm². According to the data from group 8, when both the medium-resistance and medium-high resistance values are higher, the highest surface loss of the anti-corona coating reaches 0.699 W/cm², which does not meet the anti-corona demands. Overall, the lowest surface loss is observed in group 4, with a maximum value of 0.430 W/cm². In this group, the resistivity of the medium-resistance layer is 10⁷ Ω·m, and the difference in resistivity between adjacent anti-corona coatings is one order of magnitude.

4.2. Nonlinearity Coefficient Analysis

Another important parameter affecting the electric field and loss distribution is the nonlinearity coefficient, β, which describes the rate of resistance variation. This coefficient plays a crucial role in determining the uniformity of the electric field distribution. As the electric field increases, the presence of nonlinearity helps to homogenize the electric field more effectively. The nonlinearity coefficient was adjusted to 5 cm/kV and 0.1 cm/kV, and the resulting highest electric field values were calculated. It is shown in Figure 12. The explanation of the colors is in

Table 8. Nonlinear value.

Groups	Medium Resistance Coating (cm/kV)	Medium-High Resistance Coating (cm/kV)	High Resistance Coating (cm/kV)	Color
1	0.6	0.8	1.0	Pink
2	0.7	0.9	1.1	Gray
3	0.8	1.0	1.2	Red
4	0.9	1.1	1.3	Green
5	1.0	1.2	1.4	Blue

. When a material with a higher nonlinearity coefficient is used, large surface currents flow through the anti-corona coating under a high electric field, leading to significant losses. As the electric field continues to rise, the anti-corona coating may become damaged. However, if the nonlinearity coefficient is too small, the surface current flowing through the anti-corona coating is insufficient. If the anti-corona coating does not reach the rated voltage at its end, flashover can occur. Thus, the nonlinearity coefficient cannot be too high or too low.

For SiC materials, the recommended value range is 0.50–1.50 cm/kV. In this anti-corona design, there are three segments of nonlinear anti-corona coating. According to Table 8, the nonlinear values in each resistance layer are discussed. From the analysis of the one-dimensional electric field distribution line graph, the optimal nonlinearity values can be determined.

According to the analysis and comparison, when the nonlinearity coefficient in group 3 is selected, the electric field distribution is the most uniform. At the exit slot, the sudden change in the electric field is minimal, and there is no significant variation in the electric field at the junction between the medium resistance and medium-high resistance layers. Additionally, the highest electric field value is effectively reduced to 3.51 kV/cm. The maximum electric field line chart is shown in Figure 13.

5. Conclusions

In this study, electric field concentration in the end of large hydro-generator is an urgent problem. There are still some problems for research method of resistance-capacitance chain model. The motor insulation anti-corona structure and electric field distribution cannot be a true reflection. Meanwhile, the electric field at the corner and loss distribution are unobtainable, which increases the calculation error. In this manuscript, the electric field distribution and loss in the end is accurately analyzed by the finite element method. The optimal anti-corona structure and material properties are obtained, which provides good guidance for wire bar production.

The electric field distribution of the stator bar was subjected to optimization analysis, leading to the following conclusions:

• When the corner angle was set to 22.5°, the electric field distribution was the most uniform, exhibiting the lowest maximum electric field value, which was 13.8% lower than that for the 15° corner angle;
• A sudden change in the electric field occurred at the junction between the medium resistance and medium-high resistance layers. Through comparative analysis, the optimal length distribution of the anti-corona coating was identified. For the best anti-corona effect, the three-dimensional electric field value of the anti-corona coating was identified to be approximately 2.58 kV/cm, while the corresponding one-dimensional electric field value was 4.22 kV/cm;
• The effects of the surface resistivity and nonlinearity coefficient of anti-corona materials on the electric potential and electric field distribution at the end of the wire bar were explored. Test results indicated that the surface resistivity in adjacent anti-corona segments should differ by one order of magnitude, with the medium resistance selected from the 10⁷ order of magnitude. This reduced the loss to 0.430 W/cm². By adjusting the nonlinearity coefficient, the optimal value range for the nonlinearity coefficient was identified.