Permanent magnet synchronous motor non-inductive control method of high-order sliding mode observer based on inverse hyperbolic sine function

Abstract

The invention discloses a permanent magnet synchronous motor non-inductive control method of a high-order sliding mode observer based on an inverse hyperbolic sine function. The method comprises the following steps: acquiring three-phase current and voltage of a permanent magnet synchronous motor; obtaining current and voltage under a static coordinate system after Clark conversion; subtraction is carried out on the current under the static coordinate system and the observed current obtained through observation of the high-order sliding mode observer, and a current error value is obtained; performing inverse hyperbolic sine function operation on the current error value, and combining with the observed electrical angular velocity of the rotor to obtain an extended counter electromotive force signal containing a harmonic signal; filtering the extended counter electromotive force signal containing the harmonic signal to obtain an extended counter electromotive force signal; and extracting observation values of a motor rotor position, a rotor electrical angle and an electrical angular speed in the extended back electromotive force signal through a homogenized phase-locked loop. According to the invention, the problem that the existing high-order sliding-mode observer based on the segmented sine function is poor in observation precision when observing the position of the motor rotor can be solved, and errors of the observation rotating speed and angle can be eliminated.

Description

Technical Field

[0001] This invention relates to the field of sensorless control of permanent magnet synchronous motors, and specifically to a sensorless control method for permanent magnet synchronous motors based on a high-order sliding mode observer of an inverse hyperbolic sine function. Background Technology

[0002] With the development and innovation of related power control technologies, the performance of AC speed control systems has been continuously improved, gradually gaining a dominant position in the field of motor control. Permanent magnet synchronous motors (PMSMs) replace the excitation windings that generate the magnetic field on the rotor of synchronous motors with permanent magnets, simplifying the motor mechanism and reducing production and assembly costs. Because there is no excitation current causing excitation losses, the power density and efficiency of the motor are significantly improved. In PMSM drive control, real-time detection of the motor rotor position information is an indispensable condition for achieving closed-loop vector control; therefore, the introduction of sensors is necessary. The introduction of sensors improves motor control accuracy, but, aside from other factors, the most important factor is the unavoidable increase in cost. This is not the optimal choice for applications where high motor control accuracy is not required, thus necessitating continuous optimization of sensorless control strategies.

[0003] Based on the speed range of the motor, sensorless control strategies for permanent magnet synchronous motors can be divided into three categories: composite control strategies applicable to zero-low speed domain, medium-high speed domain, and full-speed domain operation. When the motor operates in the zero-low speed domain, physical quantities related to motor speed or position information, such as flux linkage and back electromotive force, have very small amplitudes, resulting in a low signal-to-noise ratio and low energy extraction efficiency. Open-loop starting and high-frequency signal injection control strategies are typically employed. In the medium-high speed domain, estimation is mostly based on the motor model. This relies on the motor's fundamental mathematical model, using various algorithms to process the mathematical model to obtain physical quantities related to motor speed or rotor position angle, from which rotor position information is extracted. These include flux linkage estimation, extended Kalman filter algorithm, model reference adaptive method, Romberg observer, and sliding mode observer algorithm. Among these, the sliding mode observer method has low requirements for system model accuracy and is insensitive to parameter changes and external disturbances, making it a highly robust control method widely used in engineering. However, traditional sliding mode observers use a sign function as the switching function, which introduces chattering into the system. To eliminate chattering, a low-pass filter needs to be introduced. The cutoff frequency of the low-pass filter cannot be too small or too large, otherwise the observed extended back electromotive force will lag and still contain a lot of harmonics, thus reducing the accuracy of the position observation.

[0004] To address the issue of poor accuracy in observing the rotor position of a motor using traditional sliding mode observers, existing technologies have replaced traditional sliding mode observers with higher-order sliding mode observers. This involves replacing the sign function used in traditional sliding mode observers with a piecewise sine function as the switching function, which reduces chattering to some extent. However, some chattering is still retained, causing errors in the observed rotational speed and angle. Summary of the Invention

[0005] Purpose of the invention: The purpose of this invention is to provide a sensorless control method for permanent magnet synchronous motors based on a high-order sliding mode observer using an inverse hyperbolic sine function. This method can solve the problem of poor observation accuracy of the motor rotor position by existing high-order sliding mode observers based on piecewise sine functions, and eliminate errors in the observed speed and angle.

[0006] Technical solution: The present invention provides a sensorless control method for a permanent magnet synchronous motor based on a high-order sliding mode observer of an inverse hyperbolic sine function, comprising:

[0007] The natural coordinate system of the permanent magnet synchronous motor is obtained by sampling the sampling resistors of the three-phase lower bridge arm of the inverter. The three-phase current and three-phase voltage of the permanent magnet synchronous motor are obtained by Clark transformation. The current and voltage in the stationary coordinate system are the actual current and voltage values ​​of the permanent magnet synchronous motor in the stationary coordinate system.

[0008] The current and voltage in the stationary coordinate system are fed into a pre-designed high-order sliding mode observer. The difference between the current in the stationary coordinate system and the observed current obtained by the high-order sliding mode observer is used to obtain the current error value. The current error value is processed by an inverse hyperbolic sine function and combined with the observed electric angular velocity of the rotor to obtain an extended back electromotive force signal containing harmonic signals.

[0009] An adaptive complex coefficient filter is used to filter the extended back EMF signal containing harmonic signals. The filter removes the harmonic signals and obtains the filtered extended back EMF signal. The filtered extended back EMF signal is then fed back to the high-order sliding mode observer.

[0010] The rotor electrical angle and rotor electrical angular velocity observations are extracted from the filtered extended back EMF signal by a uniformized phase-locked loop. The errors between the rotor electrical angle observations and the corresponding standard values, and the errors between the rotor electrical angular velocity observations and the corresponding standard values ​​are calculated. The two errors are input into the speed loop PI and the current loop PI for modulation output. After inverse Park transformation, the required control voltage information is output. The corresponding pulse width modulation is obtained through space vector pulse width modulation and input into the permanent magnet synchronous motor to make it run stably, thereby realizing sensorless control of the permanent magnet synchronous motor.

[0011] Furthermore, the mathematical model of the permanent magnet synchronous motor is as follows:

[0012] ;

[0013] In the formula, , In the stationary coordinate system Stator voltage and current under the shaft; , They are respectively Stator voltage and current under the shaft; , These are the stator inductance and resistance, respectively. for Extended back electromotive force under the axis; for Extended back electromotive force under the axis; For magnetic linkage; It is an electrical angle; It represents the electric angular velocity.

[0014] Furthermore, the natural coordinate system of the permanent magnet synchronous motor The three-phase currents and three-phase voltages are subjected to Clark transformation to obtain the currents and voltages in the stationary coordinate system, including:

[0015] For the natural coordinate system of permanent magnet synchronous motor The three-phase currents underwent Clark transformation, and the current transformation process is as follows:

[0016] ;

[0017] in, In a stationary coordinate system Stator current under the shaft; In a stationary coordinate system Stator current under the shaft; Let A be the current in phase A in the natural coordinate system; Let B be the current in the natural coordinate system; Let be the current in phase C in the natural coordinate system.

[0018] Furthermore, the mathematical expression for the higher-order sliding mode observer is as follows:

[0019] ;

[0020] In the formula, for The observed current under the axis; for The observed current under the axis; for The difference between the observed current and the actual current under the shaft; for The difference between the observed current and the actual current under the shaft; for Extended back electromotive force observed off-axis; This represents the extended back electromotive force observed along the β axis; It is an inverse hyperbolic sine function; , All are sliding mode gain terms.

[0021] Furthermore, the expression for the inverse hyperbolic sine function is as follows:

[0022] ;

[0023] In the formula, It is a constant parameter.

[0024] Furthermore, the mathematical expression for the adaptive complex coefficient filter is as follows:

[0025] ;

[0026] In the formula, for Extended back electromotive force observed off-axis; for Extended back electromotive force observed off-axis; for Unfiltered extended back EMF under shaft; for Unfiltered extended back EMF under shaft; The center frequency; is the cutoff frequency, and s is the complex frequency variable.

[0027] Furthermore, the cutoff frequency The expression is as follows:

[0028] ;

[0029] in, This is the cutoff frequency coefficient.

[0030] Furthermore, the errors in calculating the observed rotor electrical angle and the corresponding standard value, and the errors in calculating the observed rotor electrical angular velocity and the corresponding standard value, are performed using the following formulas:

[0031] ;

[0032] in, To observe the electrical angle; The observed electric angular velocity; This represents the error increment; The center frequency; For magnetic linkage; The electrical angle of the motor; for The filtered extended back EMF signal under the shaft; for The extended back EMF signal after filtering at the shaft.

[0033] Furthermore, when hour, The error increment As the input to the PI controller, the PI controller adjusts the phase of the observed electrical angle to remain constant, thus... Phase lock is achieved.

[0034] Furthermore, the output of the PI controller is the observed electrical angular velocity of the motor. ,right The observed electrical angle is obtained by integration. .

[0035] Beneficial effects: Compared with the prior art, the significant technical effects of the present invention are as follows: (1) The present invention adopts a high-order sliding mode observer and designs the switching function as an inverse hyperbolic sine function; at the same time, it replaces the traditional low-pass filter in the extended back EMF estimation with an adaptive complex coefficient filter, and obtains the rotor position and electrical angular velocity in conjunction with a uniform phase-locked loop. The above combined technical solutions work together to address the problems of severe chattering, large filter phase lag, and insufficient phase-locked loop robustness in traditional methods, and propose a systematic improvement path. Among them, the high-order sliding mode observer, combined with the inverse hyperbolic sine switching function, reduces the chattering caused by the switching term, improves the noise resistance and convergence smoothness of the observer, thereby alleviating the position and velocity estimation fluctuation problem caused by chattering; (2) Compared with the fixed parameter low-pass filter, the adaptive complex coefficient filter significantly reduces the phase lag and amplitude attenuation caused by the amplitude frequency response, can adjust the parameters online according to the operating state, improves the extraction accuracy of the extended back EMF, and solves the problem of estimation lag and deviation caused by traditional filtering; (3) The uniform phase-locked loop is not sensitive to modeling and parameter perturbation, and can still lock stably under weak signal and sudden change of operating conditions. Compared with conventional PLL, it improves the robustness to motor parameter uncertainty and electromagnetic interference, and solves the problem of slow locking and easy loss of lock; (4) Through the above improvements, the present invention significantly improves the observation accuracy of motor rotor position and electric angular velocity, shortens the convergence and locking time under dynamic operating conditions (such as sudden change of speed), reduces estimation fluctuation and phase error, and enhances the stability and reliability under noise and parameter disturbance, thereby helping to reduce the dependence on mechanical sensors and system cost. Attached Figure Description

[0036] Figure 1 This is a schematic diagram of the process of the present invention;

[0037] Figure 2 This is a schematic diagram of the structure of a uniform phase-locked loop;

[0038] Figure 3The simulation diagram shows the rotational speed error observed by a high-order sliding mode observer based on a piecewise sine function.

[0039] Figure 4 Simulation diagram of rotational speed error observed by a high-order sliding mode observer based on an inverse hyperbolic sine function;

[0040] Figure 5 Simulation diagram of electrical angle error observed by a high-order sliding mode observer based on a piecewise sine function;

[0041] Figure 6 The simulation diagram shows the electrical angle error observed by a high-order sliding mode observer based on an inverse hyperbolic sine function. Detailed Implementation

[0042] The technical solution of the present invention will now be described in detail with reference to specific embodiments and accompanying drawings.

[0043] like Figure 1 As shown, the present invention provides a sensorless control method for a permanent magnet synchronous motor based on a high-order sliding mode observer of an inverse hyperbolic sine function, comprising the following steps:

[0044] S1. Sample the sampling resistors of the three-phase lower bridge arm of the inverter to obtain the three-phase current in the natural coordinate system ABC of the permanent magnet synchronous motor. , , and three-phase voltage , , For the three-phase currents of a permanent magnet synchronous motor in the natural coordinate system ABC , , and three-phase voltage , , Perform a Clark transformation to obtain the current in the stationary coordinate system. , and voltage in stationary coordinate system , Current in a stationary coordinate system , Voltage in stationary coordinate system , This represents the actual current and voltage values ​​of the permanent magnet synchronous motor in the stationary coordinate system.

[0045] In this embodiment, the mathematical model of the permanent magnet synchronous motor is as follows:

[0046] ;

[0047] In the formula, , In the stationary coordinate system Stator voltage and current under the shaft; , In the stationary coordinate system Stator voltage and current under the shaft; , These are the stator inductance and resistance, respectively. for Extended back electromotive force under the axis; for Extended back electromotive force under the axis; For magnetic linkage; It is an electrical angle; It represents the electric angular velocity.

[0048] In this embodiment, the natural coordinate system of the permanent magnet synchronous motor is... The three-phase currents underwent Clark transformation, and the current transformation process is as follows:

[0049] ;

[0050] in, In a stationary coordinate system Stator current under the shaft; In a stationary coordinate system Stator current under the shaft; Let A be the current in phase A in the natural coordinate system; Let B be the current in the natural coordinate system; Let be the current in phase C in the natural coordinate system.

[0051] The voltage transformation process is similar. Since this invention uses constant amplitude transformation, the coefficients before the Clark transformation matrix are 2 / 3.

[0052] S2. Substitute the current and voltage in the stationary coordinate system into a pre-designed high-order sliding mode observer. The current in the stationary coordinate system... , Observation current obtained from higher-order sliding mode observer , By subtracting the values, we obtain the current error value. , Current error value , The calculation was performed using an inverse hyperbolic sine function, combined with the observed electric angular velocity of the rotor. This yields an extended back electromotive force signal containing a large number of harmonic signals. , .

[0053] The mathematical expression for the higher-order sliding mode observer is as follows:

[0054] ;

[0055] In the formula, for The observed current under the axis; for The observed current under the axis; for The difference between the observed current and the actual current under the shaft; for The difference between the observed current and the actual current under the shaft; for Extended back electromotive force observed off-axis; for Extended back electromotive force observed off-axis; It is an inverse hyperbolic sine function; , All are sliding mode gain, and .

[0056] In this embodiment, the expression for the inverse hyperbolic sine function is as follows:

[0057] ;

[0058] In the formula, As a constant parameter, in this invention .

[0059] The inverse hyperbolic sine function has the following properties:

[0060] (1) Singularity and dissipation:

[0061] ;

[0062] when hour, .

[0063] (2) Derivative and Lipschitz constant:

[0064] ;

[0065] Therefore, globally:

[0066] ;

[0067] Take the Lipschitz constant ,Right now:

[0068] .

[0069] S3. Using an adaptive complex coefficient filter to process the extended back electromotive force signal containing harmonic signals (i.e., and The signal is filtered to remove most of the harmonic signals, resulting in the filtered extended back electromotive force signal (i.e., and The filtered extended back EMF signal is then fed back to the higher-order sliding mode observer.

[0070] In this embodiment, the mathematical expression for the adaptive complex coefficient filter is as follows:

[0071] ;

[0072] In the formula, for Extended back electromotive force observed off-axis; for Extended back electromotive force observed off-axis; for Unfiltered extended back EMF under shaft; for Unfiltered extended back EMF under shaft; The center frequency; The cutoff frequency; It is a complex frequency variable.

[0073] Expanding a complex vector can be written in matrix form:

[0074] ;

[0075] In order to make the center frequency and cutoff frequency follow the changes in motor speed, the electrical angular velocity output by the observer is fed back to the filter as its center frequency and corresponding cutoff frequency. Designed as follows:

[0076] ;

[0077] in, The cutoff frequency coefficient is used in this invention. .

[0078] S4, such as Figure 2 As shown, the filtered extended back EMF signal is extracted through a uniformized phase-locked loop. , The included rotor electrical angle observations Rotor electric angular velocity observations ; Calculate the observed rotor electrical angles Errors relative to the corresponding standard values, and observed rotor electric angular velocity values The error between the standard value and the speed loop PI and the current loop PI are input to the speed loop PI and the current loop PI for modulation output. After inverse Park transformation, the required control voltage information is output. After space vector pulse width modulation, the corresponding pulse width modulation is obtained and input to the permanent magnet synchronous motor to make it run stably, thereby realizing sensorless control of the permanent magnet synchronous motor.

[0079] In step S4, the rotor electrical angle observation value and rotor electric angular velocity observations The acquisition process is as follows:

[0080] The filtered extended back EMF signal is extracted using a homogenized phase-locked loop. , Then, using the formula right , After numerical homogenization and passing through a phase-locked loop, the observed rotor electric angular velocity values ​​are obtained. ;right Through the integral operator Obtain rotor electrical angle observation values ,

[0081] In step S4, the rotor electrical angle observation value Errors relative to the corresponding standard values, and observed rotor electric angular velocity values The error compared to the corresponding standard value is calculated using the following formula:

[0082] ;

[0083] in, To observe the electrical angle; The observed electric angular velocity; This represents the error increment; The center frequency; For magnetic linkage; The electrical angle of the motor; for The filtered extended back EMF signal under the shaft; for The extended back EMF signal after filtering at the shaft.

[0084] when hour, The error increment As the input to the PI controller, the PI controller adjusts the phase of the observed electrical angle to remain constant, thus... Phase lock is achieved.

[0085] The output of the PI controller is the observed electrical angular velocity of the motor. ,right The observed electrical angle can be obtained by integration. .

[0086] Compared with existing technologies, the present invention can effectively reduce the error of rotational speed observation and rotor position observation, filter out high-order harmonics, and improve the observation accuracy.

[0087] This invention simulates the mounting as follows Figure 1 As shown in the model, the selected motor parameters are derived from the permanent magnet synchronous motor product model 42JSF630AS-1000 manufactured by Wildfire Corporation. This invention will experimentally compare the existing high-order sliding mode observer using a piecewise sine function as the switching function with the high-order sliding mode observer based on an inverse hyperbolic sine function proposed in this invention, hereinafter referred to as models HO1 and HO2. The motor parameters are shown in the table below:

[0088] Table 1 Motor Parameters

[0089]

[0090] To facilitate the analysis of simulation results, the basic parameters of the PMSM used in the simulation remained unchanged, while the important simulation parameters were set differently. m=220, kE=5 (The acceptable value range is 0.01~1). The given speed of the motor is set to 1200 r / min, and the motor is operating under light load. The simulation results are as follows: Figure 3-6 As shown. In the simulation diagram, the labels with the suffixes -motor and -HO represent the actual rotor correlation quantities and the rotor correlation quantities estimated by the higher-order sliding mode observer, respectively. Figure 3-4 The simulation results show the rotor speed and observed speeds of models HO1 and HO2 under light load (0.05N) operating conditions of the motor. It can be observed that both models quickly and smoothly reach the target speed. Figure 4 It can be seen that the HO2 model, which uses a higher-order sliding mode observer with an inverse hyperbolic sine function, is superior to the HO1 model for rotor speed estimation. Figure 3 The values ​​shown are more accurate than the motor speed error range of ±2 r / min. Figure 5-6 This is a rotor angle error diagram, compared to Figure 5 ,Depend on Figure 6 It can be seen that the improved observation model HO2 can track the actual rotor position more quickly. Figure 6 In the error curve, the positional error is between 0.0055 rad and 0.0065 rad, which is significantly lower than that of a higher-order sliding mode observer based on a piecewise sinusoidal composite function. Figure 5 The results show a position error of 0.008 rad to 0.009 rad, which provides higher estimation accuracy.

Claims

1. A sensorless control method for a permanent magnet synchronous motor based on a high-order sliding mode observer using an inverse hyperbolic sine function, characterized in that, include: The natural coordinate system of the permanent magnet synchronous motor is obtained by sampling the sampling resistors of the three-phase lower bridge arm of the inverter. The three-phase current and three-phase voltage under the following conditions; for the natural coordinate system of the permanent magnet synchronous motor The three-phase current and three-phase voltage are measured. Transformation yields the current and voltage in the stationary coordinate system, which are the actual current and voltage values ​​of the permanent magnet synchronous motor in the stationary coordinate system. The current and voltage in the stationary coordinate system are fed into a pre-designed high-order sliding mode observer. The difference between the current in the stationary coordinate system and the observed current obtained by the high-order sliding mode observer is used to obtain the current error value. The current error value is calculated using an inverse hyperbolic sine function and combined with the observed electric angular velocity of the rotor to obtain an extended back electromotive force signal containing harmonic signals. An adaptive complex coefficient filter is used to filter the extended back EMF signal containing harmonic signals. The filter removes the harmonic signals and obtains the filtered extended back EMF signal. The filtered extended back EMF signal is then fed back to the high-order sliding mode observer. The rotor electrical angle and rotor electrical angular velocity observations contained in the filtered extended back EMF signal are extracted by a uniformized phase-locked loop. The errors of the observed rotor electrical angle and the corresponding standard value, and the errors of the observed rotor electrical angular velocity and the corresponding standard value are calculated. The two errors are input into the speed loop PI and the current loop PI for modulation output. After inverse Park transformation, the required control voltage information is output. After space vector pulse width modulation, the corresponding pulse width modulation is obtained and input into the permanent magnet synchronous motor to make it run stably, thereby realizing the sensorless control of the permanent magnet synchronous motor.

2. The sensorless control method for permanent magnet synchronous motors based on a high-order sliding mode observer using an inverse hyperbolic sine function according to claim 1, characterized in that, The mathematical model of the permanent magnet synchronous motor is as follows: ； In the formula, , These are the stator voltage and current along the α axis in the stationary coordinate system, respectively. , They are respectively Stator voltage and current under the shaft; , These are the stator inductance and resistance, respectively. for Extended back electromotive force under the axis; for Extended back electromotive force under the axis; For magnetic linkage; It is an electrical angle; It represents the electric angular velocity.

3. The sensorless control method for permanent magnet synchronous motors based on a high-order sliding mode observer using an inverse hyperbolic sine function according to claim 1, characterized in that, The natural coordinate system of the permanent magnet synchronous motor The three-phase currents and three-phase voltages are subjected to Clark transformation to obtain the currents and voltages in the stationary coordinate system, including: Clark transformation is performed on the three-phase current of the permanent magnet synchronous motor. The current transformation process is as follows: ； in, In a stationary coordinate system Stator current under the shaft; In a stationary coordinate system Stator current under the shaft; Let A be the current in phase A in the natural coordinate system; Let B be the current in the natural coordinate system; Let be the current in phase C in the natural coordinate system.

4. The sensorless control method for permanent magnet synchronous motors based on a high-order sliding mode observer using an inverse hyperbolic sine function according to claim 1, characterized in that, The mathematical expression for the higher-order sliding mode observer is as follows: ； In the formula, for The observed current under the axis; for The observed current under the axis; for The difference between the observed current and the actual current under the shaft; for The difference between the observed current and the actual current under the shaft; for Extended back electromotive force observed off-axis; for Extended back electromotive force observed off-axis; It is an inverse hyperbolic sine function; , All are sliding mode gain terms.

5. The sensorless control method for permanent magnet synchronous motors based on a high-order sliding mode observer with an inverse hyperbolic sine function according to claim 1, characterized in that, The expression for the inverse hyperbolic sine function is as follows: ； In the formula, It is a constant parameter.

6. The sensorless control method for permanent magnet synchronous motors based on a high-order sliding mode observer with an inverse hyperbolic sine function according to claim 1, characterized in that, The mathematical expression for the adaptive complex coefficient filter is as follows: ； In the formula, for Extended back electromotive force observed off-axis; for Extended back electromotive force observed off-axis; for Unfiltered extended back EMF under shaft; for Unfiltered extended back EMF under shaft; The center frequency; The cutoff frequency; It is a complex frequency variable.

7. The sensorless control method for permanent magnet synchronous motors based on a high-order sliding mode observer using an inverse hyperbolic sine function according to claim 6, characterized in that, The cutoff frequency The expression is as follows: ； in, This is the cutoff frequency coefficient.

8. The sensorless control method for permanent magnet synchronous motors based on a high-order sliding mode observer with an inverse hyperbolic sine function according to claim 1, characterized in that, The following formulas are used to calculate the errors between the observed rotor electrical angle values ​​and the corresponding standard values, and the errors between the observed rotor electrical angular velocity values ​​and the corresponding standard values: ； in, To observe the electrical angle; The observed electric angular velocity; This represents the error increment; The center frequency; For magnetic linkage; The electrical angle of the motor; for The filtered extended back EMF signal under the shaft; This is the filtered extended back EMF signal along the β axis.

9. The sensorless control method for permanent magnet synchronous motors based on a high-order sliding mode observer with an inverse hyperbolic sine function according to claim 8, characterized in that, when hour, The error increment As the input to the PI controller, the PI controller adjusts the phase of the observed electrical angle to remain constant, thus... Phase lock is achieved.

10. The sensorless control method for permanent magnet synchronous motors based on a high-order sliding mode observer with an inverse hyperbolic sine function according to claim 8, characterized in that: The output of the PI controller is the observed electrical angular velocity of the motor. ,right The observed electrical angle is obtained by integration. .

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