1 Introduction

As an important part of the power system, the safe and stable operation of large generators has always been a hot issue for many researchers. As one of the common faults of generator, the stator winding inter-turn short circuit fault of large generator can cause electromagnetic noise and radial vibration when the fault occurs. When the fault is serious, it can cause stator core damage and a series of problems [1]. Therefore, it is of practical significance to research the inter-turn short circuit of generator stator winding to ensure the safe and stable operation of the generator.

Scholars have done some research work on the inter turn short circuit fault of generator stator winding. In reference [2], the influence of stator winding inter-turn short circuit position on the electromagnetic torque fluctuation characteristics is studied when the air gap of synchronous generator is eccentric. It is pointed out that the angle between the electromagnetic torque and the position of short circuit and minimum air gap is approximately cosine. In reference [3], the influence of stator winding inter-turn short circuit on the induction electromotive force of generator is studied, and the potential difference before and after the fault is proposed to diagnose the fault. In reference [4,5,6,7], the vibration characteristics and the influencing factors of the generator after inter-turn short circuit under different working conditions are studied. In reference [8], a new detecting coil method for detecting the inter-turn short circuit fault of generator winding is proposed. In reference [9], the variation characteristics of negative sequence current component after stator winding inter-turn short circuit fault are studied. In reference [10], the field circuit coupling method is used to study the dynamic electromagnetic force distribution on the top and the wall of the rotor teeth when the stator winding turns are short circuited. In reference [11], single winding analysis and finite element analysis are used to study the characteristics of inter-turn short circuit fault of generator stator winding. It is pointed out that short circuit current, phase current, torque and radial force are related to fault location and fault degree. In reference [12, 13], the variation characteristics of air gap magnetic field and electromagnetic torque in case of rotor vibration eccentricity and stator winding inter-turn short circuit fault are studied. It shows that the second harmonic vibration changes obviously after the fault. It is proposed to detect the eccentricity of the rotor by using the 17th and 19th harmonic currents. In reference [14], the field circuit coupling method is used to study the distribution of eddy current loss on the rotor surface after inter-turn short circuit of generator stator winding.

The above references have made some achievements in the research of generator stator winding inter-turn short circuit fault. However, in most of the above references, the research on electromagnetic torque is only limited to fault identification. The change rule and influence factors of each harmonic component of electromagnetic torque after fault are not analyzed in depth. Therefore, it is necessary to study the characteristics of electromagnetic torque in the case of inter-turn short circuit fault in the stator winding of nuclear generator. Based on the above references, taking a large nuclear power half speed turbo-generator as the research object, and adopting the field circuit network coupling method, it is established that a coupled simulation model for the operation of nuclear power generator with rated load in grid. On this basis, the inter-turn short circuit fault of the stator winding is simulated. The variation characteristics of internal magnetic field and electromagnetic torque before and after fault are studied. The variation law of the components of air gap flux density and electromagnetic torque after fault is given. It is pointed out that there is a one-to-one relationship between the relative position of the fault winding to the pole and harmonic components of the electromagnetic torque. The research results provide theoretical reference for the diagnosis and location of stator winding inter-turn short circuit fault.

2 Field-Circuit-Network Coupled Mathematical Model

Taking the one machine infinite bus system as the research object, the equivalent circuit of the system is shown in Fig. 1.

Fig. 1
figure 1

Equivalent circuit of one machine infinite bus system

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2.1 Generator Magnetic Field Equation

It is assumed that the permeability and conductivity of iron core are considered as isotropic, the magnetic field of generator is quasi-static field, the displacement current and the eddy current loss of stator core are ignored. The boundary-value problem of the generator 2D transient electromagnetic field is written as

$$ \left\{ {\begin{array}{*{20}l} {\Omega :\frac{\partial }{\partial x}(\frac{1}{\mu }\frac{\partial A}{{\partial x}}) + \frac{\partial }{\partial y}(\frac{1}{\mu }\frac{\partial A}{{\partial y}}) = - J + \sigma \frac{dA}{{dt}}} \hfill \\ {\Gamma :A = 0} \hfill \\ \end{array} } \right. $$
(1)

where μ is magnetic permeability, J is source current density, σ is electrical conductivity, and A is magnetic vector potential, Γ is the boundary of the outer circle of stator and the inner circle of rotor.

2.2 Mathematical Model of Generator

2.2.1 Generator Stator Winding Loop Equation

As shown in Fig. 1, the generator stator winding loop equation can be expressed as

$$ \begin{aligned} {\varvec{U}}_{{\text{s}}} & = {\varvec{E}}_{{\text{s}}} - {\varvec{R}}_{{\text{s}}} {\varvec{I}}_{{\text{s}}} - {\varvec{L}}_{{\text{s}}} \frac{{{\text{d}}{\varvec{I}}_{{\text{s}}} }}{{{\text{d}}t}} \\ & = - l_{{{\text{ef}}}} {\varvec{C}}_{{\text{s}}}^{{\text{T}}} \frac{{\partial {\varvec{A}}}}{\partial t} - {\varvec{R}}_{{\text{s}}} {\varvec{I}}_{{\text{s}}} - {\varvec{L}}_{{\text{s}}} \frac{{{\text{d}}{\varvec{I}}_{{\text{s}}} }}{{{\text{d}}t}} \\ \end{aligned} $$
(2)

where Es is the induction electromotive force of generator, \({\varvec{E}}_{{\text{s}}} = [e_{{\text{a}}} ,e_{{\text{b}}} ,e_{{\text{c}}} ]^{{\text{T}}}\), Us and Is are the armature terminal voltage and current respectively, \({\varvec{U}}_{{\text{s}}} = [u_{{\text{A}}} ,u_{{\text{B}}} ,u_{{\text{C}}} ]^{{\text{T}}}\), \({\varvec{I}}_{{\text{s}}} = [i_{{\text{A}}} ,i_{{\text{B}}} ,i_{{\text{C}}} ]^{{\text{T}}}\), Rs and Ls are stator winding resistance matrix and end leakage inductance matrix respectively, \({\varvec{R}}_{{\text{s}}} = diag[R_{{\text{s}}} ,R_{{\text{s}}} ,R_{{\text{s}}} ]^{{\text{T}}}\), and \({\varvec{L}}_{{\text{s}}} = diag[L_{{\text{s}}} ,L_{{\text{s}}} ,L_{{\text{s}}} ]^{{\text{T}}}\), lef is the effective axial length of the generator, and Cs is the incidence matrix of stator current.

The current density of stator winding is as follows:

$$ J_{{\text{s}}} = \frac{{i_{{\text{s}}} }}{a} \cdot \frac{1}{{S_{{\text{s}}} }} $$
(3)

where is is the phase current of stator winding, a is the number of parallel branches of stator winding, and SS is the sectional area of stator winding.

2.2.2 Excitation Loop Equation

As shown in Fig. 1, the excitation loop equation can be expressed as

$$ \begin{aligned} u_{{\text{f}}} & = e_{{\text{f}}} + R_{{\text{f}}} i_{{\text{f}}} + L_{{\text{f}}} \frac{{{\text{d}}i_{{\text{f}}} }}{{{\text{d}}t}} \\ & = - l_{{{\text{ef}}}} {\varvec{C}}_{{\text{f}}}^{{\text{T}}} \frac{{\partial {\varvec{A}}}}{\partial t} + R_{{\text{f}}} i_{f} + L_{{\text{f}}} \frac{{{\text{d}}i_{{\text{f}}} }}{{{\text{d}}t}} \\ \end{aligned} $$
(4)

where uf is excitation voltage, ef is induced electromotive force of excitation winding, if is excitation current, Rf and Lf are excitation winding resistance and end leakage inductance respectively, and Cf is the correlation matrix of excitation current.

The current density of excitation winding is as follows:

$$ J_{{\text{f}}} = {{i_{{\text{f}}} } \mathord{\left/ {\vphantom {{i_{{\text{f}}} } {s_{{\text{f}}} }}} \right. \kern-\nulldelimiterspace} {s_{{\text{f}}} }} $$
(5)

where Sf is the sectional area of excitation winding.

2.2.3 Damping Loop Equation

The schematic diagram of damping circuit is shown in Fig. 2. In the figure, Rdi and Ldi are the resistance and leakage inductance of the damping end ring respectively, Rzi and Lzi are the resistance and inductance of the damping winding, Rwi and Lwi are the slot wedge resistance and inductance, ibi is the sum of the damping bar and slot wedge current; idi is the loop current; udi is the voltage at both ends of the damping bar.

Fig. 2
figure 2

Schematic diagram of damping circuit

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Suppose that Jci and Jzi are the current density of the straight line part of the wedge and the damping bar respectively, then

$$ \left\{ {\begin{array}{*{20}c} {J_{ci} = - \sigma_{1} \frac{{\partial {\varvec{A}}}}{\partial t} + \sigma_{1} \frac{{u_{{{\text{d}}i}} }}{{l_{{{\text{ef}}}} }}} \\ {J_{zi} = - \sigma_{2} \frac{{\partial {\varvec{A}}}}{\partial t} + \sigma_{2} \frac{{u_{{{\text{d}}i}} }}{{l_{{{\text{ef}}}} }}} \\ \end{array} } \right. $$
(6)

where σ1 and σ2 are the electrical conductivity of the slot wedge and damping bar respectively.

The damping loop equation can be expressed as [15]

$$ \left\{ \begin{gathered} {\varvec{C}}_{{{\text{d1}}}} {\varvec{U}}_{{\text{d}}} + {\varvec{C}}_{{{\text{d2}}}} {\varvec{I}}_{d} + {\varvec{D}}_{{{\text{d1}}}} \frac{\partial A}{{\partial t}} = 0 \hfill \\ {\varvec{C}}_{{{\text{d3}}}} {\varvec{U}}_{{\text{d}}} + {\varvec{C}}_{{{\text{d4}}}} {\varvec{I}}_{{\text{d}}} + {\varvec{D}}_{{{\text{d2}}}} \frac{{{\text{d}}I_{{\text{d}}} }}{{{\text{d}}t}} = 0 \hfill \\ \end{gathered} \right. $$
(7)

2.2.4 Rotor Core Eddy Current

For the hydro generator, the rotor is composed of laminated form, so the eddy current effects can be neglected. For turbo generator, its rotor is solid, therefore, it can be treated by the method of modifying conductivity of iron core linear part [16].

$$ J_{{\text{R}}} = - \sigma_{{\text{R}}} \frac{{\partial {\varvec{A}}}}{\partial t} $$
(8)

where σR is the rotor material conductivity.

2.3 Mathematical Model of Power Grid

2.3.1 Transformer Equation

The transformer which is connected to the generator adopts Dyn11 connection form. As shown in Fig. 1, the voltage and the current equations of transformer can be written as

$$ {\varvec{E}}_{{\text{t}}} = k{\varvec{G}}_{1} \left( {{\varvec{U}}_{{\text{s}}} - {\varvec{R}}_{{\text{t}}} {\varvec{I}}_{{\text{s}}} - {\varvec{L}}_{{\text{t}}} \frac{{{\text{d}}{\varvec{I}}_{{\text{s}}} }}{{{\text{d}}t}}} \right) $$
(9)
$$ {\varvec{I}}_{{\text{t}}} = k{\varvec{G}}_{2} {\varvec{I}}_{{\text{t}}} $$
(10)

where Et is voltage on HV side of transformer, It is current on HV side of transformer, k is the transformation ratio,\({\varvec{E}}_{{\text{t}}} = [e_{{{\text{YA}}}} ,e_{{{\text{YB}}}} ,e_{{{\text{YC}}}} ]^{{\text{T}}}\), \({\varvec{I}}_{{\text{t}}} = [i_{{{\text{YA}}}} ,i_{{{\text{YB}}}} ,i_{{{\text{YC}}}} ]^{{\text{T}}}\), Rt and Lt are transformer winding resistance matrix and end leakage inductance matrix respectively,\({\varvec{R}}_{{\text{t}}} = diag[R_{{\text{t}}} ,R_{{\text{t}}} ,R_{{\text{t}}} ]\), \({\varvec{L}}_{{\text{t}}} = diag[L_{{\text{t}}} ,L_{{\text{t}}} ,L_{{\text{t}}} ]\),\({\varvec{G}}_{1} = \left[ {\begin{array}{*{20}c} 1 & 0 & { - 1} \\ { - 1} & 1 & 0 \\ 0 & { - 1} & 1 \\ \end{array} } \right]\), and \({\varvec{G}}_{2} = \left[ {\begin{array}{*{20}c} 1 & { - 1} & 0 \\ 0 & 1 & { - 1} \\ { - 1} & 0 & 1 \\ \end{array} } \right]\).

2.3.2 Transmission Line Voltage and Current Equations

As shown in Fig. 1, the transmission line voltage and current equations can be written as

$$ \left\{ {\begin{aligned} {{\varvec{E}}_{t} = {\varvec{U}} + {\varvec{R}}_{1} {\varvec{I}}_{1} + {\varvec{L}}_{1} \frac{{{\text{d}}{\varvec{I}}_{1} }}{{{\text{d}}t}}} \\ { = {\varvec{U}} + {\varvec{R}}_{2} {\varvec{I}}_{2} + {\varvec{L}}_{2} \frac{{{\text{d}}{\varvec{I}}_{2} }}{{{\text{d}}t}}} \\ \end{aligned} } \right. $$
(11)
$$ {\varvec{I}}_{\text{L}} = {\varvec{I}}_{1} + {\varvec{I}}_{2} $$
(12)

where U and IL are the infinite system voltage and current respectively, \({\varvec{U}} = [u_{{{\text{SA}}}} ,u_{{{\text{SB}}}} ,u_{{{\text{SC}}}} ]^{{\text{T}}}\), \({\varvec{I}}_{\text{L}} = [i_{{1\text{A}}} + i_{{2\text{A}}} ,i_{{1\text{B}}} + i_{{2\text{B}}} ,i_{{1\text{C}}} + i_{{2\text{C}}} ]^{\text{T}}\), R1, R2, L1 and L2 are their diagonal matrices respectively. If the transmission line is a single circuit transmission line in the system, then I2 = 0.

Bringing Eqs. (3), (5), (6) and (8) into Eq. (1), after finite element discrimination, then

$$ {\varvec{KA}} + {\varvec{C}}_{s} {\varvec{I}}_{s} + {\varvec{C}}_{f} {\varvec{I}}_{f} + {\varvec{C}}_{d5} {\varvec{U}}_{d} + {\varvec{D}}_{d3} \frac{{\partial {\varvec{A}}}}{\partial t} = 0 $$
(13)

Coupling Eqs. (2), (4), (7), (9)–(13), and eliminating the intermediate variables, the finite element equation can be got.

$$ \begin{gathered} \left[ {\begin{array}{*{20}c} {\varvec{K}} & {{\varvec{C}}_{{\text{s}}} } & {{\varvec{C}}_{{\text{f}}} } & {{\varvec{C}}_{{{\text{d5}}}} } & 0 & 0 & 0 \\ 0 & 0 & {{\varvec{R}}_{{\text{f}}} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{\varvec{C}}_{{{\text{d1}}}} } & {{\varvec{C}}_{{{\text{d2}}}} } & 0 & 0 \\ 0 & 0 & 0 & {{\varvec{C}}_{{{\text{d3}}}} } & {{\varvec{C}}_{{{\text{d4}}}} } & 0 & 0 \\ 0 & { - k{\varvec{G}}_{1} ({\varvec{R}}_{{\text{s}}} + {\varvec{R}}_{{\text{t}}} )} & 0 & 0 & 0 & { - {\varvec{R}}_{1} } & 0 \\ 0 & { - k{\varvec{G}}_{1} ({\varvec{R}}_{{\text{s}}} + {\varvec{R}}_{{\text{t}}} )} & 0 & 0 & 0 & 0 & { - {\varvec{R}}_{2} } \\ 0 & { - {\varvec{M}}} & 0 & 0 & 0 & {k{\varvec{G}}_{2} } & {k{\varvec{G}}_{2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varvec{A}} \\ {{\varvec{I}}_{{\text{s}}} } \\ {{\varvec{I}}_{{\text{f}}} } \\ {{\varvec{U}}_{{\text{d}}} } \\ {{\varvec{I}}_{{\text{d}}} } \\ {{\varvec{I}}_{1} } \\ {{\varvec{I}}_{2} } \\ \end{array} } \right] + \hfill \\ \left[ {\begin{array}{*{20}c} {{\varvec{D}}_{{{\text{d3}}}} } & 0 & 0 & 0 & 0 & 0 & 0 \\ { - l_{{{\text{ef}}}} {\varvec{C}}_{{\text{f}}}^{{\text{T}}} } & 0 & {{\varvec{L}}_{{\text{f}}} } & 0 & 0 & 0 & 0 \\ {{\varvec{D}}_{{{\text{d1}}}} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\varvec{D}}_{{{\text{d2}}}} } & 0 & 0 \\ { - kl_{{{\text{ef}}}} {\varvec{G}}_{1} {\varvec{C}}_{{\text{s}}}^{{\text{T}}} } & { - k{\varvec{G}}_{1} ({\varvec{L}}_{{\text{s}}} + {\varvec{L}}_{{\text{t}}} )} & 0 & 0 & 0 & { - {\varvec{L}}_{1} } & 0 \\ { - kl_{{{\text{ef}}}} {\varvec{G}}_{1} {\varvec{C}}_{{\text{s}}}^{{\text{T}}} } & { - kG_{1} ({\varvec{L}}_{{\text{s}}} + {\varvec{L}}_{{\text{t}}} )} & 0 & 0 & 0 & 0 & { - {\varvec{L}}_{2} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\frac{\partial }{\partial t}\left[ {\begin{array}{*{20}c} {\varvec{A}} \\ {{\varvec{I}}_{{\text{s}}} } \\ {{\varvec{I}}_{{\text{f}}} } \\ {{\varvec{U}}_{{\text{d}}} } \\ {{\varvec{I}}_{{\text{d}}} } \\ {{\varvec{I}}_{1} } \\ {{\varvec{I}}_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ {u_{{\text{f}}} } \\ 0 \\ 0 \\ {\varvec{U}} \\ {\varvec{U}} \\ 0 \\ \end{array} } \right] \hfill \\ \end{gathered} $$
(14)

where M = diag[1, 1, 1].

3 Simulation and Experiment

In this section, taking the MJF-30-6 non-salient pole synchronous generator in the national laboratory as an example, the simulation model is built. In the simulation model, the synchronous generator is connected with the infinite system through the transformer and the double circuit transmission line. When the generator is stable operation with 1/2 rated load, it is simulated that phase A winding first branch of generator occurs 2% to 22% inter-turn short circuit fault in 3 s. The fault is removed after 0.1 s, then, the generator returns to normal operation.

At the same time, in order to verify the correctness of the simulation model, the related experiment is carried out. In the experiment, the MJF-30-6 non-salient pole synchronous generator is connected with 11 kV grid through the transformer. The voltage step-up ratio of transformer is 2.5. And the connection group of transformer is Dyn11. In the experiment, the LM-1 excitation device and F6-10,000 special recording wave analysis device for experiment are used. The experiment site is shown in Fig. 3. And the simulation and experiment results are shown in Fig. 4.

Fig. 3
figure 3

Experiment site

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Fig. 4
figure 4

Comparison between stator current simulation result and experimental result

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As seen from Fig. 4, the simulation result of stator winding current is in agreement with the experiment result. The correctness of the simulation model is verified.

4 Harmonic Analysis of Air Gap Magnetic Field in Stator Winding with Inter-turn Short Circuit

In order to study the characteristics of inter-turn short circuit in stator winding of nuclear generator, the field-circuit-network coupled simulation model is established based on the case of a 1407MVA nuclear half speed turbo-generator connected to the infinite system with rated load. In this model, the generator is connected with the infinite system through the transformer and the double circuit transmission line. The generator is modeled by finite element method, and the influence of structure and magnetic saturation are fully considered in the modeling process. Main parameters of generator are shown in Table 1:

Table 1 Main parameters of nuclear power half speed turbine generator
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In order to improve the accuracy of the calculation, the layered mesh generation method is adopted in the air gap mesh generation. The meshing result is shown in Fig. 5.

Fig. 5
figure 5

Nuclear power generator finite element mesh cross-section figure

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Under the condition of stable operation of nuclear power generator with rated load, 0–25% inter-turn short circuit fault in the first branch of stator phase A winding is simulated. The simulation results of air gap flux density before and after the fault are shown in Fig. 6.

Fig. 6
figure 6

Air gap flux density distribution before and after stator winding inter-turn short circuit fault

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It can be seen from Fig. 6 that after the fault, the air gap flux density of the generator affected by the short-circuit current is no longer symmetrical, and the air gap flux density of the area where the short-circuit winding is located has obvious distortion.

The air gap flux density of the generator before and after the fault is decomposed by FFT, and the result is shown in Fig. 7.

Fig. 7
figure 7

Frequency spectrum of air gap flux density before and after stator winding inter-turn short circuit fault

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From the spectrum analysis results in Fig. 7, it can be seen that the air gap flux density only contains fundamental and odd harmonic components during normal operation of the generator. After the inter-turn short circuit fault, the air gap flux density of generator not only contains fundamental and odd harmonic components, but also even harmonic components.

5 Analysis of the Influence Factors of the Harmonic Component in the Air Gap Magnetic Field During the Inter-Turn Short Circuit of the Stator Winding

In order to study the influence factors of air gap flux density harmonic component after inter-turn short circuit fault of stator winding, In this section, the characteristics of air gap flux density change when the relative position of stator and rotor changes and the characteristics of air gap flux density change when the relative position of stator and rotor is constant are studied.

5.1 Influence of Relative Position Between Stator and Rotor on Air Gap Flux Density

In order to study the change characteristics of air gap magnetic density when the relative position of stator fault winding axis and magnetic pole axis changes with the same degree of inter-turn short circuit, The simulation of 0–25%inter-turn short circuit of the first branch winding of phase A is carried out. It is assumed that the relative position angle between the pole axis and the stator fault winding axis is α, Due to limited space, in this paper, the distribution of the magnetic field line of the generator is given only when the magnetic pole axis is 9° ahead of the fault winding axis (α = 9°). As it is shown in Fig. 8.

Fig. 8
figure 8

Distribution of magnetic force line after stator winding inter-turn short circuit fault

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When the relative position of magnetic pole axis and fault winding axis changes under the stator winding inter-turn short circuit fault, the partial results of air gap magnetic density distribution are shown in Fig. 9.

Fig. 9
figure 9

Air gap magnetic density with change of α

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As can be seen from Fig. 9, with the change of relative position angle α, the degree of air gap magnetic flux density distortion also changes.

The harmonic analysis of the air gap flux density at different α after the fault is carried out, and the results are shown in Fig. 10.

Fig. 10
figure 10

Frequency spectrum of air gap flux density at different α

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It can be seen from Fig. 10 that the distribution of the components of the air gap magnetic density has certain regularity with the change of α.

When the magnetic field in the area where the fault winding is located is the strongest, the fundamental component of air gap flux density is the smallest and the harmonic component is the largest.

5.2 Influence of Different Degree of Fault on Air Gap Flux Density

When the relative position between the axis of stator fault winding and the axis of magnetic pole is constant, the simulation of stator winding inter-turn short circuit fault in different degrees is carried out. The simulation results of air gap flux density in different degrees of fault are shown in Fig. 11.

Fig. 11
figure 11

Air gap magnetic density distribution in different degree of fault

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The analysis results of air gap flux density spectrum under different degrees of fault are shown in Fig. 12. It can be seen from Fig. 12 that when the relative position between the axis of stator fault winding and the axis of magnetic pole is constant, The fundamental component of air gap flux density decreases with the increase of fault degree, while the harmonic component of air gap flux density increases with the increase of fault degree.

Fig. 12
figure 12

Frequency spectrum of air gap flux density in different degree of fault

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6 Analysis of Electromagnetic Torque in Stator Winding with Inter-turn Short Circuit

In order to study the harmonic components and influencing factors of the electromagnetic torque after the inter-turn short circuit fault in the stator winding of nuclear power generator, the analysis of the electromagnetic torque in the case of 0–37.5% inter-turn short circuit fault is carried out. The results of electromagnetic torque spectrum analysis are shown in Fig. 13.

Fig. 13
figure 13

Frequency spectrum of electromagnetic torque of generators after fault

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From the spectrum analysis results in Fig. 13, it can be seen that the electromagnetic torque of nuclear power generator after stator winding inter-turn short circuit fault contains not only the DC component in normal operation, but also the even harmonic component.

6.1 Influence of Different Degree of Fault on Electromagnetic Torque

The harmonic analysis of the electromagnetic torque under normal operation and different degrees of faults of the nuclear generator is shown in Fig. 14.

Fig. 14
figure 14

Frequency spectrum of electromagnetic torque of generators before and after fault

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From the harmonic analysis results in Fig. 14, it can be seen that during fault operation, the electromagnetic torque generates even harmonic component in addition to DC component. In addition, with the increase of fault degree, the harmonic components of electromagnetic torque also increase. The DC component of the electromagnetic torque after the fault is obviously smaller than that of the normal operation, and the more serious the fault is, the smaller the DC component of the electromagnetic torque is.

6.2 Influence of Relative Position of Stator and Rotor on Electromagnetic Torque

An analysis is made of the electromagnetic torque when the relative position of the axis of the fault winding and the axis of the magnetic pole changes after the inter-turn short circuit fault of the stator winding. The results are shown in Fig. 15.

Fig. 15
figure 15

Electromagnetic torque of α change after fault

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According to the simulation results in Fig. 15, when α = 27°, the electromagnetic torque is the minimum. And this characteristic does not change with the degree of failure. It can be seen from Fig. 8 that the position of fault winding is exactly the maximum position of air gap Synthetic magnetic field. That is, the larger the air-gap synthetic magnetic field at the fault winding location, the smaller the electromagnetic torque. Therefore, this feature can be used to assist in determining the location of the faulty winding.

7 Conclusions

In this paper, the air gap magnetic field and electromagnetic torque of the generator in the case of an inter-turn short circuit in the stator windings of a nuclear power half-speed turbo-generator are studied. The main conclusions are as follows:

  1. (1)

    When the inter-turn short circuit occurs in the stator winding of a nuclear power half speed turbo-generator, the air gap flux density does not only contain the odd harmonic component in normal operation, but also the even harmonic component. At the same time, the change in air gap flux density is related to the relative position of the axis of the fault winding and the axis of the magnetic pole. The stronger the magnetic field in the area where the faulty winding is located, the smaller the fundamental component of the air-gap flux density and the larger the harmonic component. When the relative position of the fault winding axis and the magnetic pole axis is constant, the more serious the fault is, the more serious the magnetic field distortion, the smaller the fundamental component of the air-gap flux density, and the larger the harmonic component.

  2. (2)

    When the inter-turn short circuit fault of stator winding occurs, in addition to the DC component, the electromagnetic torque also generates an even harmonic component. And the DC component decreases with the increase of fault degree, while the harmonic component increases with the increase of fault degree. The larger the air gap synthetic magnetic field at the location of the fault winding, the smaller the electromagnetic torque. And this characteristic does not change with the fault degree.

Based on the existing research results, this paper further analyzes the change characteristics and influencing factors of the air gap magnetic field and electromagnetic torque after the inter-turn short circuit of the stator windings of a nuclear power generator. It is of certain significance to realize the fault diagnosis and location using the electromagnetic torque after the fault.