Permanent magnet synchronous motor turn-to-turn short circuit fault diagnosis and positioning method

Abstract

The invention discloses a permanent magnet synchronous motor turn-to-turn short circuit fault diagnosis and positioning method. According to the method, extra hardware is not needed, whether the turn-to-turn short circuit fault occurs in the motor can be accurately diagnosed only based on the voltage and current signals of the two-phase stator coordinate system (alpha-beta coordinate system), the severity of the fault is quantitatively represented, and the fault phase is accurately positioned. The method does not need an accurate model or a large amount of data, is not affected by environmental interference, is accurate in positioning, is simple to implement, and is high in real-time performance.

Description

Technical Field

[0001] The present invention relates to the technical field of permanent magnet motor fault diagnosis, and in particular to a method for diagnosing and locating an inter-turn short circuit fault of a permanent magnet synchronous motor. Background Art

[0002] Permanent magnet synchronous motors (PMSMs) are widely used in electric vehicles, rail transit, aerospace, and other fields due to their high efficiency, high power density, and high torque density. However, high load operation and the high switching frequencies of wide-bandgap devices increase mechanical, thermal, and electrical stresses, leading to frequent inter-turn short circuit faults (ITSFs). Failure to diagnose these faults can lead to performance degradation or system downtime, resulting in economic losses. Therefore, efficient and accurate fault diagnosis methods are crucial for reliable motor operation.

[0003] Currently, motor interturn short-circuit fault diagnosis primarily involves three approaches: mathematical model-based, data-driven, and signal analysis. Mathematical model-based diagnosis relies on accurately modeling the motor and comparing the differences between expected and actual operation. Theoretically, if a precise theoretical model of the motor can be constructed, this difference analysis can effectively locate interturn short-circuit faults. However, this approach is limited by the accuracy of the motor model. Variations in motor parameters with load and temperature during operation can easily lead to model deviations, affecting diagnostic accuracy. In contrast, data-driven approaches do not rely on precise motor physical models. Instead, they leverage extensive data from permanent magnet synchronous motors under various fault levels and operating conditions. Machine learning or data mining algorithms are used to uncover underlying patterns and regularities within the data, thereby mapping data features to fault type and severity. This approach relies on both data quality and quantity, and as a black-box model, it is difficult to interpret diagnostic evidence. Signal analysis-based diagnosis utilizes the time and frequency domain characteristics of electrical, magnetic, mechanical, and thermal signals, analyzing the motor's characteristic signals through mathematical transformations to determine if a fault has occurred. This method does not require a large amount of data and accurate models, but the electromagnetic interference in the motor operating environment and the sensor's own error drift will affect the extraction of fault characteristic signals, resulting in misjudgment or missed faults. In addition, the fault location accuracy is limited, and it is difficult to accurately determine the specific fault location. Especially in the case of minor faults in multi-phase windings or complex motor structures, it is often necessary to combine other methods for assistance, which increases the complexity and cost of diagnosis. Summary of the Invention

[0004] In light of this, the present invention provides a method for diagnosing and locating inter-turn short-circuit faults in permanent magnet synchronous motors. This method, based solely on voltage and current signals in a two-phase stator coordinate system (α-β coordinate system), requires no additional hardware and can accurately diagnose whether a motor has experienced an inter-turn short-circuit fault, quantitatively characterize the fault severity, and precisely locate the fault phase. This method does not require precise models or large amounts of data, is unaffected by environmental interference, and offers precise positioning, ease of implementation, and high real-time performance.

[0005] The method for diagnosing an inter-turn short circuit fault of a permanent magnet synchronous motor of the present invention comprises:

[0006] S1, samples the three-phase current of the motor and converts it to the stator two-phase coordinate system to obtain i α ,i β ;

[0007] S2, extraction i α ,i β The fundamental wave i α1 ,i β1 ;

[0008] S3, i α1 Delay 90° and get i α1 e -j90° ;

[0009] S4, calculate the fault factor FI, the fault factor FI is: i α1 e -j90° with i β1 The absolute value of the difference in one current cycle T s the average of the inner integral;

[0010] If the fault factor FI is greater than the set threshold Th, it is determined that a turn-to-turn short circuit fault has occurred.

[0011] Preferably, in S4, the integration operation is implemented by a low-pass filter, and its transfer function is 1 / (Ts+1), where s is the Laplace operator and T is the integration time constant.

[0012] Preferably, in S4, the motor current frequency is extracted and its reciprocal is taken to obtain the period T S .

[0013] Preferably, the motor current frequency is obtained by converting the actual motor speed, performing a phase-locked loop operation on the motor current signal, or performing a Fourier transform on the motor current signal and selecting the frequency component with the largest amplitude.

[0014] Preferably, in S4, the threshold Th is determined in the following manner:

[0015] Run the motor when there is no fault, take multiple sets of data to calculate FI, and calculate the mean value μ of FIFI Standard deviation σ FI , calculate the initial threshold TH0 = μ FI +kσ FI , where k is 2 to 3;

[0016] According to motor parameters, operating conditions, minimum short-circuit fault turns ratio μ min and the three-phase permanent magnet motor voltage equation to estimate the minimum fault I fmin ; According to the fault factors FI and I fmin The relationship between FI=4μI f / 3πCalculate the FI at the minimum fault min ;

[0017] The set threshold TH is greater than the initial threshold TH0 and less than the minimum fault FI min .

[0018] Preferably, the severity of the inter-turn short circuit fault is determined according to the size of the fault factor FI: the larger the fault factor FI is, the more serious the inter-turn short circuit fault is.

[0019] Preferably, in S2, the fundamental wave is extracted through a bandpass filter, a signal phase-locked loop or a wavelet transform.

[0020] The present invention also provides a method for locating an inter-turn short-circuit fault of a permanent magnet synchronous motor, comprising:

[0021] S-1, sample the dq axis voltage of the motor and convert it to the stator two-phase coordinate system to obtain u α ,u β ;

[0022] S-2, extract u α ,u β The fundamental wave u α1 ,u β1 , and delay the phase by 90° to obtain u α1 e -j90° ,u β1 e -j90° ;

[0023] S-3, will u α1 e -j90° and u α1 Convert to DC model to get u1, u2; β1 e -j90° and u β1 Convert to DC model to get u3, u4;

[0024] S-4, calculate the failure factors FI1, FI2, and FI3 as follows:

[0025]

[0026] And take the signs of the fault factors to obtain sign(FI1), sign(FI2), sign(FI3);

[0027] S-5, judge based on the sign of the fault factor:

[0028] When sign(FI1) is greater than 0 and sign(FI2) is less than 0, a turn-to-turn short circuit fault occurs in phase A;

[0029] When sign(FI2) is greater than 0 and sign(FI3) is greater than 0, a turn-to-turn short circuit fault occurs in phase B;

[0030] When sign(FI1) is less than 0 and sign(FI3) is less than 0, an inter-turn short circuit fault occurs in phase C.

[0031] Preferably, in S-2, the fundamental wave is extracted by a bandpass filter, a signal phase-locked loop or a wavelet transform.

[0032] Preferably, in S-2, an all-pass filter or Hilbert transform is used to achieve a 90° delay of the voltage signal.

[0033] Beneficial effects:

[0034] 1. The required current signal in the fault diagnosis method of the present invention can be obtained by the current sensor of a conventional motor driver. No additional observer is required. Only the current signal of the two-phase stator coordinate system (α-β coordinate system) is used to diagnose whether a short-circuit fault has occurred, thereby improving versatility. In signal processing, first, the Clark is transformed into the α-β coordinate system, and then the α-axis current is lagged by 90 degrees to extract the fault characteristics. By comparing with the preset threshold, the fault can be accurately diagnosed and the degree of the fault can be determined. The size of the diagnostic sensitivity is related to the preset threshold. In the specific implementation: the signal processing is simple, and part of the signal processing is a necessary part of the motor control algorithm implementation, so the calculation amount is lower and it has more efficient real-time performance; the absence of additional sensors also reduces hardware costs and interference effects; the sensitivity of the method can also be adjusted by adjusting the size of the preset threshold.

[0035] 2. Based on diagnosis, the present invention generates a DC signal by delaying the voltage signal output by the current controller by 90 degrees and performing Park transformation, defines fault factors (FI1, FI2, FI3) and accurately locates the phase of the motor short circuit fault according to their polarity combination. The signal noise output by the controller is lower, which has less impact on phase judgment. The calculation and judgment rules are simple, which effectively improves the real-time performance of the system.

[0036] 3. The present invention integrates diagnosis and positioning processes, integrates diagnosis and positioning through a unified signal processing process, and shares the signal acquisition and processing module with the normal drive control part of the motor, which simplifies the system design. It can also locate the fault phase while diagnosing the fault. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 Schematic diagram of the inter-turn short circuit of phase A of the permanent magnet synchronous motor.

[0038] Figure 2 Flowchart for diagnosis of inter-turn short circuit fault in permanent magnet synchronous motor.

[0039] Figure 3 Flowchart for locating inter-turn short-circuit fault in permanent magnet synchronous motor. DETAILED DESCRIPTION

[0040] The present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0041] The present invention provides a method for diagnosing and locating an inter-turn short-circuit fault of a permanent magnet synchronous motor.

[0042] 1. Diagnosis of inter-turn short circuit fault

[0043] The present invention first analyzes the equivalent circuit model of a permanent magnet synchronous motor when a turn-to-turn short circuit occurs. Taking the turn-to-turn short circuit of phase A as an example, the equivalent circuit diagram is as follows: Figure 1 As shown in Figure 1, the A-phase winding is divided into a healthy part and a faulty part. The resistance, inductance and back electromotive force of the healthy part are represented by R ah , L ah and e ah The resistance, inductance and back electromotive force of the fault part are represented by R af , L af and e af . R f is the contact resistance of the inter-turn short circuit fault, which is used to characterize the degree of insulation degradation. f The smaller it is, the greater the degree of insulation damage. f is the contact resistance R f The current in the branch. R x , L x , e x are the resistance, inductance and back EMF of motor phase x respectively. The short-circuit turns ratio is defined as μ=N c / N s , which indicates the number of turns with inter-turn short-circuit fault (N c ) and the total number of turns of the motor phase winding (N s ), the larger μ is, the deeper the short-circuit fault is.

[0044] When a short circuit occurs in phase A of the motor, the voltage equation in the stator two-phase coordinate system can be expressed as:

[0045]

[0046] Among them, u α ,u β They are the motor α-β axis voltage, R s is the phase resistance of the motor, i α ,i β are the motor α-β axis currents, L d is the inductance of the motor's direct axis, θ e is the electrical angle of the motor, ω e is the electrical angular velocity of the motor, ψ f is the permanent magnet flux, i f is the motor fault current, which is expressed as:

[0047]

[0048] Fault current i introduced by inter-turn short circuit fault f This will cause current distortion in the α-β plane. When there is no fault, the α-β axis currents are sinusoidal currents with equal amplitude and orthogonal to each other:

[0049]

[0050] Among them, i q is the motor q-axis current.

[0051] After a short circuit occurs between turns of phase A, the α-β axis current is:

[0052]

[0053] Among them, i αh and i βh is the α-β plane current in the fault-free state, i αf and i βf is the α-β plane current after the a-phase inter-turn short circuit fault occurs. It can be seen that when the motor A phase inter-turn short circuit fault occurs, the α axis current is equivalent to the superposition of (-2μ / 3)i in the healthy mode. f The fault current is reduced, while the β-axis current maintains the current value under no-fault conditions.

[0054] Consider only the fault current i f The fundamental component of the fault current is expressed as:

[0055] i f =-I f sin(θ e +θ k )

[0056] Among them, I f is the fault current i f The amplitude, θ k (k=a,b,c) is the phase of the fault current.

[0057] Through the above analysis, the α-β plane current under the A phase inter-turn short circuit fault can be expressed as

[0058]

[0059] Since the α-β axis current is a sinusoidal signal, it is difficult to directly extract the fault characteristics. We can design an all-pass filter, Hilbert transform, etc. to delay the α axis current phase by 90 degrees, so that it is in phase with the β axis current. Then, the α axis current with a phase delay of 90 degrees is subtracted from the β axis current to extract the relevant fault current I f fault characteristics.

[0060] will i α The phase lag is 90°, giving:

[0061]

[0062] Among them, i α e -j90° This is the result of a 90° delay in the current phase of the α-axis. α e -j90° with i β In the same phase, the fault current I can be extracted by subtracting the two. f The fault characteristics are shown in the following formula:

[0063]

[0064] To intuitively characterize the severity of the fault, the absolute value of the above fault characteristics is taken and integrated and averaged within a current cycle to obtain the fault factor FI:

[0065]

[0066] The integral term in the above equation will amplify the low-frequency noise in the sampled current, causing some interference with the result. Therefore, a low-pass filter (transfer function is 1 / (Ts+1)) is used instead of the integral (transfer function is 1 / s), where T is the time constant, which is related to the bandwidth of the low-pass filter. It can be set accordingly according to the actual motor operating conditions to achieve a good integration effect without excessively amplifying the low-frequency noise of the current.

[0067] On the other hand, during the operation of the motor, the time of one current cycle T S It may not be possible to determine it exactly, but it can be estimated based on the motor current frequency.S The main methods are: (1) Using the relationship between motor speed and current frequency, measure the actual speed of the motor, convert it to the motor current frequency, and take the reciprocal to get the period T S (2) The motor current signal can be used to perform phase-locked loop operation, extract the current frequency, and take the inverse to obtain the period T S (3) The motor current signal can be Fourier transformed, and the frequency component with the largest amplitude can be selected to obtain the fundamental frequency of the current. The reciprocal is taken to obtain the period T S .

[0068] The fault factor is only related to the short-circuit turns ratio and the magnitude of the fault current, where T s The above analysis is based on the example of a turn-to-turn short circuit fault in phase A. Due to the symmetry of the three phases of the motor, it is easy to obtain that when a turn-to-turn short circuit fault occurs in phases B and C, i α e -j90° -i β The value of the fault factor FI is 4μI after the absolute value integration and averaging. f / 3π. This shows that no matter which phase of the three-phase permanent magnet motor has a turn-to-turn short circuit fault, as long as the fault severity is the same, that is, μ and R f The fault factor FI will show the same value. Therefore, the fault factor FI can be used as a fault indicator to diagnose whether the motor has a turn-to-turn short circuit fault and determine the severity of the motor fault.

[0069] In actual judgment, a threshold value, TH, can be preset and the calculated FI can be compared with TH for judgment. However, the value of TH determines the sensitivity of this method. When TH is too high, the sensitivity is low, and minor faults may be missed. When TH is too low, the sensitivity is high, and the method is easily affected by signal noise, resulting in false fault alarms.

[0070] The goal of setting TH is to distinguish between normal operation (no fault, FI≈0) and fault state (FI>0), while balancing the false alarm rate and missed alarm rate. The following are the specific methods and considerations for setting TH:

[0071] 1. First, when there is no fault, μ and I f Theoretically, FI is 0 at this time, but in practice, due to noise, sensor error and system non-ideality, FI may not be 0. You can run the motor when there is no fault, calculate FI using multiple sets of data, and calculate the mean value of FI μ FI Standard deviation σ FI , set the initial threshold TH0 = μ FI +kσ FI, where k can be selected as 2 to 3 (for a confidence interval of 95% to 99%) to avoid false positives.

[0072] 2. Determine the minimum fault level that TH can detect (i.e., the minimum μ and I f First, estimate the minimum fault I according to the motor parameters and operating conditions. f , calculate the corresponding minimum fault FI min .

[0073] In summary, set TH slightly higher than the initial threshold TH0, but lower than the minimum fault FI min At the same time, the operating results of the motor under different working conditions can be appropriately adjusted based on the above.

[0074] 2. Phase location of inter-turn short circuit fault

[0075] In order to locate the fault phase, the present invention uses the α-β plane voltage signal for analysis. When there is no fault, the α-β plane voltage signal is a sinusoidal signal with equal amplitude and orthogonal to each other. The corresponding α-β plane voltage expression is

[0076]

[0077] in, θ e is the electrical angle of the motor; u d 、u q are the voltages of the motor’s direct and quadrature axes respectively; is the phase of the voltage.

[0078] According to the above formula, the α-β plane voltages when a turn-to-turn short circuit fault occurs on phases A, B, and C can be expressed as follows:

[0079] Phase A short circuit fault:

[0080]

[0081] Phase B short circuit fault:

[0082]

[0083] Phase C short circuit fault:

[0084]

[0085] in, R s is the motor stator resistance, ω e is the electrical angular velocity of the motor, L d is the inductance of the motor quadrature axis.

[0086] Because the α-β plane voltages are orthogonal sinusoidal quantities, they cannot be directly processed and used to extract fault features. After a 90° phase delay, the resulting variable remains sinusoidal and must be converted to a DC quantity to extract fault features and distinguish between the different phases of a short-circuit fault. Delaying the voltage signal by 90° can be achieved by designing an all-pass filter and employing a Hilbert transform.

[0087] Furthermore, when a turn-to-turn short circuit fault occurs on phases A, B, and C, u α and u α e -j90° ,u β and u β e -j90° Perform Park transformation as shown below.

[0088]

[0089] The corresponding results of u1, u2, u3, and u4 when short circuit faults occur in different phases are shown in Table 1.

[0090] Table 1

[0091]

[0092]

[0093] The failure factor is defined as shown below.

[0094]

[0095] The corresponding calculated fault factors FI1 FI2 FI3 are shown in Table 2

[0096] Table 2

[0097]

[0098] It is easy to see from the results in Table 2 that the phase where the motor inter-turn short circuit occurs can be located according to the polarity of FI1, FI2, and FI3. When a short circuit fault occurs in phase A, FI1 is positive and FI2 is negative; when a short circuit fault occurs in phase B, FI2 and FI3 are positive; when a short circuit fault occurs in phase C, FI1 and FI3 are negative.

[0099] 3. Implementation steps

[0100] 3.1 Fault diagnosis process (see Figure 2 )

[0101] Step 1: Sample the three-phase current of the motor and obtain the two-phase stator current information i through Clark transformation. α ,i β .

[0102] Step 2, extract i α ,i β The fundamental signal is i α1 ,i β1 .

[0103] Since the sampled current includes not only the fundamental signal but also the interference of higher harmonics, if the sampled three-phase current signal contains too much interference, the i α ,i β Errors may occur, affecting the accuracy of fault factor calculations. The calculation of the fault factor is related to the fundamental current signal, while higher harmonics or other irrelevant signal components are often irrelevant to the fault signature and may even obscure fault information. By extracting the fundamental signal, we can directly focus on the signal components most relevant to the fault, significantly reducing these interferences and ensuring signal processing accuracy. Methods for extracting the signal fundamental include designing a bandpass filter near the fundamental, designing a phase-locked loop (PLL), and using wavelet transforms.

[0104] Step 3: Delay the current of the α axis by 90 degrees to obtain i α1 e -j90° , if a short circuit occurs, i α1 e -j90° After subtracting the β-axis current, we should get:

[0105]

[0106] Step 3: Since it is still a sinusoidal AC signal, for the convenience of comparison, take the absolute value of the above formula and integrate it to obtain the fault factor FI:

[0107]

[0108] Step 4: When a short circuit occurs, the integral of the above formula should be 4μI f / 3π, where μ represents the degree of short-circuit fault. Therefore, a threshold TH can be set and the result of the above integral can be compared with TH to determine whether a short-circuit fault occurs.

[0109] 3.2 Fault Location Process (See Figure 3 )

[0110] Step 1: Take out the dq axis voltage of the motor and perform inverse park transformation to obtain the voltage signal u in the two-phase stator coordinate system. α ,u β .

[0111] Step 2: Extract the fundamental signal to obtain u α1 ,u β1 , the two-phase fundamental voltage signal is obtained as uα1 ,u β1 , respectively delayed by 90 degrees to obtain u α1 e -j90° and u β1 e -j90° .

[0112] The core of fault phase location lies in converting the α-β plane voltage signal into a DC signal through the Park transform. The fault phase is then determined based on the polarity of the fault factors (FI1, FI2, and FI3). If the input signal contains high-frequency noise or harmonic components, the Park transform result may no longer be a stable DC signal, but rather a fluctuating or distorted signal, which interferes with the determination of the polarity of the fault factors. Extracting the fundamental signal ensures high stability of the transformed signal, thereby improving the accuracy of fault factor calculation and fault phase location. Methods for extracting the fundamental signal include designing a bandpass filter near the fundamental, designing a phase-locked loop (PLL), and using wavelet transforms.

[0113] Step 3: respectively convert the two groups of AC signals with a 90° difference between each other: u α1 e -j90° and u α1 ,u β1 e -j90° and u β1 Perform park transformation to convert to DC model to obtain u1, u2, u3, and u4.

[0114] Step 4: Calculate the failure factors FI1, FI2, and FI3 as follows:

[0115]

[0116] And take the signs of the fault factors to obtain sign(FI1), sign(FI2), sign(FI3).

[0117] Step 5, make a judgment based on the sign of the fault factor: when sign(FI1) is greater than 0 and when sign(FI2) is less than 0, a turn-to-turn short circuit fault occurs in phase A; otherwise, when sign(FI3) is greater than 0, a turn-to-turn short circuit fault occurs in phase B; otherwise, a turn-to-turn short circuit fault occurs in phase C.

[0118] In summary, the above are only preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for diagnosing a turn-to-turn short-circuit fault in a permanent magnet synchronous motor, characterized in that: include: S1, samples the three-phase current of the motor and converts it to the stator two-phase coordinate system to obtain i α ,i β ; S2, extraction i α ,i β The fundamental wave i α1 ,i β1 ; S3, i α1 Delay 90° and get i α1 e -j90° ; S4, calculate the fault factor FI, the fault factor FI is: i α1 e -j90° with i β1 The absolute value of the difference in one current cycle T s the average of the inner integral; If the fault factor FI is greater than the set threshold Th, it is determined that a turn-to-turn short circuit fault has occurred.

2. The method according to claim 1, wherein In S4, the integration operation is implemented by a low-pass filter, and its transfer function is 1 / (Ts+1), where s is the Laplace operator and T is the integration time constant.

3. The method according to claim 1 or 2, wherein: In S4, the motor current frequency is extracted and its reciprocal is taken to obtain the period T S .

4. The method according to claim 3, wherein The motor current frequency is obtained by converting the actual speed of the motor, performing a phase-locked loop operation on the motor current signal, or performing a Fourier transform on the motor current signal and selecting the frequency component with the largest amplitude.

5. The method according to claim 1, wherein In S4, the threshold Th is determined in the following manner: Run the motor when there is no fault, take multiple sets of data to calculate FI, and calculate the mean value μ of FI FI Standard deviation σ FI , calculate the initial threshold TH0 = μ FI +kσ FI , where k is 2 to 3; According to motor parameters, operating conditions, minimum short-circuit fault turns ratio μ min and the three-phase permanent magnet motor voltage equation to estimate the minimum fault I fmin ; According to the fault factors FI and I fmin The relationship between FI=4μI f / 3πCalculate the FI at the minimum fault min ; The set threshold TH is greater than the initial threshold TH0 and less than the minimum fault FI min .

6. The method according to claim 1, wherein The severity of the inter-turn short circuit fault is determined according to the size of the fault factor FI: the larger the fault factor FI is, the more serious the inter-turn short circuit fault is.

7. The method according to claim 1, wherein In S2, the fundamental wave is extracted through a bandpass filter, a signal phase-locked loop or a wavelet transform.

8. A method for locating a turn-to-turn short-circuit fault in a permanent magnet synchronous motor, characterized in that: include: S-1, sample the dq axis voltage of the motor and convert it to the stator two-phase coordinate system to obtain u α ,u β ; S-2, extract u α ,u β The fundamental wave u α1 ,u β1 , and delay the phase by 90° to obtain u α1 e -j90° ,u β1 e -j90° ; S-3, will u α1 e -j90° and u α1 Convert to DC model to get u1, u2; β1 e -j90° and u β1 Convert to DC model to get u3, u4; S-4, calculate the failure factors FI1, FI2, and FI3 as follows: And take the signs of the fault factors to obtain sign(FI1), sign(FI2), sign(FI3); S-5, judge based on the sign of the fault factor: When sign(FI1) is greater than 0 and sign(FI2) is less than 0, a turn-to-turn short circuit fault occurs in phase A; When sign(FI2) is greater than 0 and sign(FI3) is greater than 0, a turn-to-turn short circuit fault occurs in phase B; When sign(FI1) is less than 0 and sign(FI3) is less than 0, an inter-turn short circuit fault occurs in phase C.

9. The method according to claim 8, wherein In the above-mentioned S-2, the fundamental wave is extracted by using a bandpass filter, a signal phase-locked loop or a wavelet transform.

10. The method according to claim 8 or 9, characterized in that In the S-2, an all-pass filter or Hilbert transform is used to delay the voltage signal by 90°.

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