Method for estimating initial position of multi-stage motor based on sinusoidal modulation of excitation signal

Abstract

The application relates to a multi-stage motor initial position estimation method based on excitation signal sine modulation. An excitation voltage is taken as a carrier signal, and after being subjected to low-frequency sine modulation, the excitation voltage is applied to the stator winding of an exciter, and a three-phase low-frequency component is induced on the stator winding of a main motor to obtain a corresponding d-axis current. The d-axis current variance will reach a maximum value at a corresponding actual d-axis angle to obtain a rotor initial position theta 0_est1 , and the polarity of theta 0_est1 is identified according to the size of the d-axis transformed current before and after power-off. The method does not involve other complex filtering, phase-locked loop and other treatments, and thus has low requirements on the processor performance. Meanwhile, secondary identification of the magnetic pole is not needed. Finally, since a low-frequency signal is taken as a modulation wave, and a low-frequency induced signal is finally used as a data source for rotor initial position estimation, the attenuation of the low-frequency induced signal by the damping winding, the exciter winding and the main motor excitation winding is small, and thus the method has a high signal-to-noise ratio.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of electric machines, and relates to a multi-stage electric machine initial position estimation method based on sinusoidal modulation of an excitation signal. Specifically, the application relates to a multi-stage brushless electrically-excited synchronous machine rotor initial position estimation method based on sinusoidal modulation of an excitation signal, which does not depend on the saliency of the electric machine, does not require secondary identification of the magnetic poles, and is not affected by the characteristics of the main machine damping winding and the excitation frequency characteristics of the exciter. BACKGROUND

[0002] Three-stage brushless synchronous machines have the advantages of good power generation quality and high reliability, and have been widely used in aircraft power systems as generators. The three-stage brushless synchronous machine is operated in the electric state to drive the aircraft engine to start, and after the start is completed, the engine drives the machine to operate in the power generation state to supply power to the on-board electrical equipment, that is, the start and power generation integration of the three-stage brushless synchronous machine is realized, the engine dedicated starter can be omitted, the system size and weight are reduced, and the integration level is improved, which is of great significance to the aircraft power system.

[0003] When the three-stage brushless synchronous machine is operated in the electric state to drive the aircraft engine to start, accurate rotor position information is required. The traditional mechanical position sensor for obtaining the rotor position has the problems of low reliability, increased system size and weight, etc. Online estimation of the rotor position of the three-stage brushless synchronous machine can omit the mechanical position sensor, realize start control without position sensor, improve system reliability, and reduce size and weight. The accuracy of the rotor initial position directly affects the size of the current at the start moment and even the success of the start, so the estimation of the rotor initial position is very important.

[0004] The three-stage brushless synchronous machine is composed of a permanent magnet auxiliary exciter, an exciter and a main machine installed coaxially, and a structural diagram thereof is shown in Figure 1 In some applications, the permanent magnet auxiliary exciter can be omitted to form a two-stage brushless synchronous machine. Since the core components and operating principles are basically the same, the three-stage brushless synchronous machine and the two-stage brushless synchronous machine are collectively referred to as a multi-stage brushless synchronous machine.

[0005] The traditional rotor initial position estimation at zero speed static state mainly utilizes the saliency or magnetic saturation characteristics of the motor, and injects an auxiliary voltage signal, extracts the response signal containing the rotor position information, and estimates the rotor position. The method of utilizing the saliency of the motor for rotor initial position estimation at zero speed static state also includes identification of the direction of the rotor magnetic pole, and the steps mainly include: 1, rotating or high-frequency signal injection; 2, band-pass filter for extracting the response signal; 3, signal demodulation; 4, low-pass filter for obtaining the signal envelope; 5, phase-locked loop or arctangent method for estimating the rotor position; 6, secondary identification of the polarity of the magnetic pole is needed for initial rotor position estimation. The steps of the method utilizing the magnetic saturation characteristics mainly include: 1, in the range of 0-2π, fixed arcuate interval pulse voltage is injected into the main stator winding, and the response current is collected and stored at the end of the pulse voltage injection; 2, the stored data is fitted and smoothed; 3, the maximum value point corresponding to the angle position is selected as the initial position of the rotor. This method has a large amount of data processing, and the estimation accuracy is greatly affected by the interval arcuate of the two pulse injections and the current pulse response detection accuracy. In addition, the methods based on main motor stator high-frequency signal injection-excitation machine stator signal detection and excitation machine stator high-frequency signal injection-main motor stator signal detection have also been studied, but these methods based on high-frequency signal injection not only have a complex processing process, but also face some common problems with the method based on the saliency of the motor, that is, they do not consider the high-frequency filtering characteristics of the damping winding, the excitation machine winding and the main motor excitation winding, which limits the application, and when these factors are considered, the signal-to-noise ratio is seriously attenuated, and the risk of failure of initial position estimation is faced.

[0006] As can be seen from the above, in the traditional method for identifying the initial position of the rotor, the following shortcomings exist: 1, the data operation and storage amount used by the method is large, and the processor requirement is high; 2, the direction of the magnetic pole needs to be identified twice in the zero speed static state; 3, when considering the high-frequency filtering characteristics of the damping winding, the excitation machine winding and the main motor excitation winding, the signal-to-noise ratio is seriously attenuated, and the risk of failure of initial position estimation is faced. SUMMARY

[0007] Technical problems to be solved

[0008] In order to avoid the shortcomings of the prior art, the present application proposes a multi-stage motor initial position estimation method based on sinusoidal modulation of excitation signals, which overcomes the shortcomings of the existing rotor initial position estimation technology, such as dependence on saliency, large data processing amount, and the need for secondary identification of the magnetic pole, especially considering the high-frequency filtering characteristics of the damping winding, the excitation machine winding and the main motor excitation winding.

[0009] Technical scheme

[0010] A method for estimating the initial position of a multi-stage motor based on sinusoidal modulation of excitation signal is characterized in that: the multi-stage brushless electrically excited synchronous motor includes a main motor and a two-phase exciter mounted coaxially; the rotor position refers to the rotor position of the main motor in the multi-stage brushless electrically excited synchronous motor; the estimation steps are as follows:

[0011] Step 1: Apply two-phase symmetrical excitation voltages modulated by a low-frequency sinusoidal modulation signal to the stator windings of the two-phase exciter, wherein the period of the low-frequency sinusoidal modulation signal is T. L The excitation carrier voltage period is T esf And T esf <T L ;

[0012] The symmetrical excitation voltage modulated by the low-frequency square wave modulation signal will induce the fundamental current and high-order harmonic current with the same excitation frequency in the three-phase winding of the exciter rotor. After the three-phase winding current passes through the rotating rectifier, it forms the excitation winding current of the main motor. Due to the low-frequency modulation effect, the high-order harmonics are filtered out, and the low-frequency pulsation will induce current in the d-axis of the stator winding of the main motor.

[0013] Step 2: After the main motor excitation current stabilizes, at fixed intervals T... sample Real-time acquisition of the three-phase induced current of the main motor stator connected to the three-phase inverter. The three-phase induced currents are denoted as I. MMA I MMB and I MMC Then a low-frequency square wave period T L The number of sampling points within the time period is n1 = T L / 2 / T sample The set of three-phase currents corresponding to each sampling point is denoted as I1=[I MMA_1 ,I MMB_1 ,I MMC_1 ],I2=[I MMA_2 ,I MMB_2 ,I MMC_2 ...I n1 =[I MMA_n1 ,I MMB_n1 ,I MMC_n1 ];

[0014] Step 3: Set the virtual angle used during the transformation to θ. n = nΔθ, 0≤n<n2-1 and n is an integer, where n2=π / Δθ, Δθ is set arbitrarily, and its size is related to the estimation accuracy and estimation time;

[0015] Step 4: Starting from n=0, θ0=0*Δθ, convert the three-phase induced currents I1, I2…I in group n1 to n1. n1 Transform each data point to a virtual d-axis and denote the transformed data set as I. d0 =[Id0_1 ,I d0_2 ...I d0_n1 After n = 1, θ1 = 1 * Δθ, the three-phase induced currents I1, I2…I in group n1 are... n1 Transform each data point to a virtual d-axis and denote the transformed data set as I. d1 =[I d1_1 ,I d1_ 2...I d1_n1 ], ... Finally, n = n² - 1, θ n2-1 = (n2-1)*Δθ, which represents the three-phase induced currents I1, I2…I in group n1. n1 Transform each data point to the virtual d-axis, and denote the transformed data as follows:

[0016] Step 5: Calculate I for each set of data d0 , The variance, denoted as v d0 ,

[0017] Step 6: Find v d0 , The maximum value v in dk k∈[0, n²-1], then θ 0est1 =k*Δθ is the initial position of the rotor without magnetic polarity;

[0018] Step 7: Disconnect the two-phase symmetrical excitation voltage modulated by a low-frequency square wave signal applied to the stator windings of the two-phase exciter mentioned in Step 1, and record the three-phase induced current of the main motor stator at two acquisition times before and after disconnection, respectively, and record them as two sets of data I. MMA1 I MMB1 I MMC1 and I MMA2 I MMB2 and I MMC2 ;

[0019] Step 8: Use θ from step six 0est1 =k*Δθ is the transformation angle, which represents the two sets of currents I before and after disconnecting the symmetrical excitation voltage in step seven. MMA1 I MMB1 I MMC1 and I MMA2 I MMB2 and I MMC2 Transform to the d-axis and denote them as I. MMd1 and I MMd2 And according to the following rules, for θ without magnetic polarity 0est1 Processing:

[0020] If I MMd1 <I MMd2, then θ 0est1 =θ 0est1 ,

[0021] If I MMd1 >I MMd2 , then θ 0est1 =θ 0est1 +π;

[0022] Step 9: The final estimated initial position of the main motor is θ. 0est =θ 0est1 .

[0023] The expression for the symmetrical excitation voltage modulated by the low-frequency sinusoidal modulation signal in the two phases is:

[0024]

[0025] Among them, U es This is the maximum voltage value. The excitation carrier voltage angular velocity, ω represents the angular velocity of the low-frequency sinusoidal modulated signal, and t represents time.

[0026] The expression for the fundamental current induced in the three-phase windings of the exciter rotor by the symmetrical excitation voltage modulated by the two-phase low-frequency sinusoidal modulation signal is as follows:

[0027]

[0028] Among them, I esr This is the maximum value of the induced current. This refers to the signal transmission phase delay caused by inductive reactance.

[0029] The expression for the main motor excitation winding current after Fourier analysis is as follows:

[0030]

[0031] in, The DC component of the excitation carrier voltage signal after passing through a rotating rectifier. The DC component of the low-frequency modulated signal after passing through a rotating rectifier. This represents the amplitude of the harmonic current signal in the main motor due to the low-frequency modulation signal. To account for the phase delay of the harmonic current signal in the main motor due to the low-frequency modulation signal, I ωes_6v v = 2, 3... represents the amplitude of the harmonic current signal of the excitation carrier voltage signal in the main motor. The phase delay of the excitation carrier voltage signal in the harmonic current signal of the main motor is denoted as .

[0032] The expression for the induced current in the d-axis of the main motor stator winding is:

[0033]

[0034] where I dc_MM is the amplitude of the induced current.

[0035] The expression of the symmetrical field voltage modulated by the two-phase low-frequency sinusoidal modulation signal is also written as:

[0036]

[0037] At this time, T L / 2 in step 2 becomes T L / 6, and the processes of steps 1 to 7 remain unchanged.

[0038] Advantages

[0039] The application provides a multi-stage motor initial position estimation method based on sinusoidal modulation of a field signal. The field voltage is used as a carrier signal, and the carrier signal is subjected to low-frequency sinusoidal modulation and then applied to the stator winding of the exciter. Due to the large inductance low-pass filtering characteristics of the exciter and the main motor field winding, the harmonic components of the six times field voltage carrier signal on the main motor field winding are filtered out, leaving only the two times low-frequency modulation signal frequency and low-order harmonic components. The three-phase low-frequency components are induced on the main motor stator winding through the mutual inductance between the main motor stator and rotor, and the current fluctuation corresponding to the actual d-axis when the three-phase low-frequency components are converted to the actual d-axis will be maximum, and the current fluctuation of the actual q-axis will be minimum. Therefore, the position information interval for the rotation conversion is fixed and increased by an angle, and when the corresponding d-axis current is obtained, the d-axis current variance will change with the position angle and reach a maximum value at the corresponding actual d-axis angle, thereby obtaining the rotor initial position θ 0_est1 . Finally, θ 0_est1 is used for the conversion of the three-phase induction current to the d-axis current before and after the excitation power-off, and the polarity of θ 0_est1 is identified according to the size of the d-axis converted current before and after the power-off. The method only needs to perform simple mathematical processing and logical operations such as equal-interval sampling of the three-phase induction current, variance calculation, and maximum value calculation, and does not involve other complex filtering, phase-locked loop processing, etc. Therefore, the performance requirement of the processor is low. At the same time, the magnetic pole does not need to be identified twice. Finally, since the low-frequency signal is used as the modulation wave, and the low-frequency induction signal is finally used as the data source for the rotor initial position estimation, the damping winding, the exciter winding, and the main motor field winding have small attenuation to the low-frequency induction signal, so that the signal-to-noise ratio is high.

[0040] The application has the following advantages:

[0041] 1) Traditional rotor initial position estimation method relying on the saliency of the main motor, after injecting a rotating high-frequency signal on the stator side of the main motor, a filter is used to extract the signal, then multiplied by the signal of the same injection frequency to demodulate, then sent to the phase-locked loop to obtain the rotor initial position without magnetic polarity, finally an additional pulse is needed to identify the magnetic polarity twice, this process not only has complex parameter matching, but also has high requirements for the performance of the controller, the present application only needs to modulate the excitation voltage signal at low frequency, apply it to the stator winding of the exciter, and then collect the induced current on the stator winding of the main motor, according to the steps shown, find the maximum variance, and according to the size relationship of the current on the d-axis after the two sampling values before and after cutting off the excitation, the initial position of the rotor of the main motor can be obtained at one time, without complex parameter matching, and the performance requirement of the processor is also lower;

[0042] 2) The traditional rotor initial position estimation method based on magnetic saturation characteristics needs to inject signals into the stator winding of the main motor at fixed intervals, and store the response current, then process the stored data by fitting, and estimate the initial position of the rotor by finding the maximum value, this method requires high storage and calculation capacity of the processor, the storage space required by the method shown in the present application is extremely limited, and no numerical fitting is required;

[0043] 3) The present application applies the low-frequency modulated excitation voltage to the stator winding of the exciter, the required effective signal is less affected by the exciter, the main motor excitation winding and the damping winding, and the signal-to-noise ratio is higher;

[0044] 4) In actual measurement, the rotor of the main motor is adjusted every 10°, and the estimation error under different actual positions is within 3°, which can meet the requirement of 10° error in the field;

[0045] 5) In actual measurement, based on the high-frequency injection method, the frequencies are 1200Hz and 400Hz respectively, and the signal attenuation is analyzed according to the proportion of the occupied direct current signal, the signal attenuation reaches 12.4dB and 10.3dB respectively, while the signal attenuation of the method shown in the present application can be ignored, which shows a high signal-to-noise ratio. BRIEF DESCRIPTION OF DRAWINGS

[0046] Figure 1 It is a schematic diagram of the three-stage brushless synchronous motor structure with damping winding.

[0047] Figure 2 It is a flow chart of the rotor initial position estimation method of the multi-stage brushless synchronous motor proposed in the present application.

[0048] Figure 3 It is the low-frequency modulated excitation voltage signal of the present application.

[0049] Figure 4The d-axis current variation trend diagram of the main motor without magnetic pole polarity when the excitation is disconnected at different actual positions of the application.

[0050] Figure 5 The virtual angle θ of the application n The d-axis and q-axis current fluctuation diagram when d increases from 0 to π step by step.

[0051] Figure 6 The d-axis current variation trend diagram of the main motor without magnetic pole polarity when the excitation is disconnected at different actual positions of the application.

[0052] Figure 7 The estimated error of the application at different actual positions. DETAILED DESCRIPTION

[0053] The application will be further described in combination with embodiments and drawings:

[0054] A multi-stage motor initial position estimation method based on sinusoidal modulation of excitation signals, characterized in that it is applied to a multi-stage brushless electric excitation synchronous motor, which includes a main motor and a two-phase excitation motor coaxially installed, and the multi-stage brushless electric excitation synchronous motor rotor position refers to the rotor position of the main motor in the multi-stage brushless electric excitation synchronous motor.

[0055] In the embodiment, the excitation motor of the multi-stage brushless synchronous motor is a two-phase excitation motor, i.e., the excitation motor stator excitation winding is a two-phase winding with a mutual phase difference of 90 electrical degrees. The excitation motor has 6 pairs of poles, and the main motor has 3 pairs of poles. The following will take the actual position of the main motor as and for detailed description.

[0056] 1. Apply a symmetric excitation voltage modulated by a two-phase low-frequency sinusoidal modulation signal to the two-phase winding (denoted as α-phase winding and β-phase winding, respectively) of the excitation motor stator, wherein the period of the low-frequency sinusoidal modulation signal is T L = 100 ms, the excitation carrier voltage period is T esf = 5 ms, the excitation carrier voltage amplitude U es = 50 V, and the low-frequency sinusoidal modulation signal amplitude is 1, then:

[0057]

[0058] 2. After the main motor excitation current stabilizes, real-time acquisition of the main motor stator three-phase induction current connected to the three-phase inverter is performed at a fixed time interval T sample = 0.5 ms, and the three-phase induction current is denoted as I MMA , I MMB and I MMC , then the sampling point number within one low-frequency square wave period T L is n1 = T L / 2 / T sample =100, and the set of three-phase currents corresponding to each sampling point is denoted as I1 = [I MMA_1 ,I MMB_1 ,I MMC_1 ],I2=[I MMA_2 ,I MMB_2 ,I MMC_2 ...I 100 =[I MMA_100 ,I MMB_100 ,I MMC_100 ];

[0059] 3. Set Δθ=pi / 60, n2=π / Δθ=60;

[0060] 4. Starting from n = 0, θ0 = 0 * Δθ, set n1 = 100 sets of three-phase induced currents I1, I2…I 100 Transform each data point to a virtual d-axis and denote the transformed data set as I. d0 =[I d0_1 ,I d0_2 ...I d0_100 After n = 1, θ1 = 1 * Δθ, n1 = 100 sets of three-phase induced currents I1, I2…I 100 Transform each data point to a virtual d-axis and denote the transformed data set as I. d1 =[I d1_1 ,I d1_2 ...I d1_100 ], ... Finally, n = n² - 1 = 59, θ 59 =59*Δθ, which will be the number of three-phase induced currents I1, I2…I1 = 100. n1 Transform each data point to a virtual d-axis and denote the transformed data set as I. d59 =[I d59_1 ,I d59_2 ...I d59_100 The transformation formula used to convert the three-phase induced current to the virtual d-axis is:

[0061]

[0062] 5. Calculate I for each set of data. d0 I d1 …I d59 The variance, denoted as v d0 v d1 …v d59 ;

[0063] 6. Sort and search v d0 v d1 …v d59 The maximum value v in dkk∈[0, n²-1], when the actual initial positions are respectively and When, find the largest element that is v d40 and v d10 ,but and These are the initial positions of the rotor without magnetic polarity;

[0064] 7. Disconnect the two-phase symmetrical excitation voltage modulated by a low-frequency sinusoidal signal applied to the stator windings of the two-phase exciter mentioned in step 1, and record the three-phase induced current of the main motor stator at two acquisition times before and after disconnection, denoted as I. MMA1 I MMB1 I MMC1 and I MMA2 I MMB2 and I MMC2 ;

[0065] 8. When At that time, the estimated position without magnetic polarity is used. To change the angle, the two sets of currents I before and after disconnecting the symmetrical excitation voltage in step 7 are... MMA1 I MMB1 I MMC1 and I MMA2 I MMB2 and I MMC2 Transform to the d-axis and denote them as I. MMd1 and I MMd2 And there is I MMd1 <I MMd2 ,but

[0066] when At that time, the estimated position without magnetic polarity is used. To change the angle, disconnect the two sets of currents I before and after the symmetrical excitation voltage in step 7. MMA1 I MMB1 I MMC1 and I MMA2 I MMB2 and I MMC2 Transform to the d-axis and denote them as I. MMd1 and I MMd2 And there is I MMd1 >I MMd2 ,but

[0067] 9: The estimated final initial position values ​​of the main motor are as follows: and

Claims

1. A method for estimating the initial position of a multi-stage motor based on sinusoidal modulation of an excitation signal, characterized in that: The multi-stage brushless electrically excited synchronous motor comprises a main motor and a two-phase excitation motor coaxially installed, the rotor position refers to the rotor position of the main motor in the multi-stage brushless electrically excited synchronous motor, and the estimation steps are as follows: Step 1: applying on the two-phase exciter stator winding a symmetrical excitation voltage modulated by a two-phase low-frequency sinusoidal modulation signal, wherein the period of the low-frequency sinusoidal modulation signal is , the excitation carrier voltage period is , and ; The symmetrical excitation voltage modulated by the two-phase low-frequency sinusoidal modulation signal will induce the same fundamental wave current and high-order harmonic current as the excitation frequency in the three-phase winding of the excitation motor rotor, the three-phase winding current passes through the rotating rectifier to form the excitation winding current of the main motor, due to the low-frequency modulation, the high-order harmonic is filtered out, and the low-frequency pulsation will induce the current in the d-axis of the main motor stator winding; Step 2: After the main motor excitation current is stable, interval fixed time Real-time acquisition of the main motor stator three-phase induction current connected with the three-phase inverter, the three-phase induction current is respectively recorded as , and , a low-frequency sinusoidal cycle Time sampling points are , and each sampling point corresponds to a set of three-phase currents respectively recorded as , … ; Step 3: Set the virtual angle used in the transformation to and integer, where , free setting, the size of which is related to the estimation accuracy and estimation time; Step 4: From start, the set of three-phase induction currents , … are transformed to the virtual axis, respectively, and the transformed data is denoted as , thereafter , the set of three-phase induction currents , … are transformed to the virtual axis, respectively, and the transformed data is denoted as , and finally , the set of three-phase induction currents , … are transformed to the virtual axis, respectively, and the transformed data is denoted as ; Step 5: Calculate the variance of each set of data , … and denote it as , … ; Step 6: Find the maximum value in , … Then is the initial position of the rotor without the polarity of the magnetic pole.​ Step 7: disconnect the symmetrical excitation voltage modulated by the two-phase low-frequency sinusoidal modulation signal applied on the two-phase excitation machine stator winding described in step 1, and record the main motor stator three-phase induction current at two acquisition time points before and after the disconnection, respectively recorded as two groups of data , , and , and ; Step 8: Use the result of Step 6 To transform the angle, transform the two sets of currents before and after the broken-symmetry excitation voltage in Step 7 , , and , and to the axis, and call them and respectively, and process without the polarity of the magnetic pole according to the following rules: If then , If then ; Step 9: The final initial position estimate of the main motor is .

2. The method of claim 1, wherein the method further comprises: The expression of the symmetrical excitation voltage modulated by the two-phase low-frequency sinusoidal modulation signal is: (1) wherein is the voltage maximum, is the excitation carrier voltage angular velocity, is the angular velocity of the low frequency sinusoidal modulation signal, is the time.

3. The method of claim 1, wherein the method further comprises: determining the initial position of the rotor of the multi-stage motor based on the sinusoidal modulation of the excitation signal. The expression of the fundamental wave current induced in the three-phase winding of the excitation motor rotor by the symmetrical excitation voltage modulated by the two-phase low-frequency sinusoidal modulation signal is: (2) wherein is the maximum value of the induced current, is the phase delay of the signal transmission caused by the inductive reactance, is the angular velocity of the excitation carrier voltage, is the angular velocity of the low-frequency sinusoidal modulation signal.

4. The method of claim 1, wherein the method further comprises: The expression of the excitation winding current of the main motor after Fourier analysis is: (3) wherein, is the DC component of the field carrier voltage signal after the rotary rectifier, is the DC component of the low frequency modulation signal after the rotary rectifier, is the harmonic current signal amplitude of the low frequency modulation signal in the main motor, is the harmonic current signal phase delay of the low frequency modulation signal in the main motor, is the harmonic current signal amplitude of the field carrier voltage signal in the main motor, is the harmonic current signal phase delay of the field carrier voltage signal in the main motor, is the angular velocity of the field carrier voltage, is the angular velocity of the low frequency sinusoidal modulation signal.

5. The method of claim 4, wherein the method further comprises: The expression of the current induced in the d-axis of the main motor stator winding is: (4) wherein is the induced current amplitude.

6. The method of claim 1, wherein the method further comprises: The expression of the symmetrical excitation voltage modulated by the two-phase low-frequency sinusoidal modulation signal is also written as: The expression of the symmetrical excitation voltage modulated by the two-phase low-frequency sinusoidal modulation signal is also written as: (5) At this time, steps 2 to 7 remain unchanged; wherein: becomes , steps 1 to 7 remain unchanged; wherein: is the angular speed of the excitation carrier voltage, is the angular speed of the low-frequency sinusoidal modulation signal, is the voltage maximum.

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