CN103501149B - A kind of multiphase induction motor specifies subharmonic electric current suppressing method - Google Patents
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Abstract
The present invention relates to a kind of multiphase induction motor and specify subharmonic electric current suppressing method, comprise the following steps: A, current reference value process; B, current feedback value process; C, current setting; D, voltage reference value are determined. It is based on multiphase induction motor vector controlled, by heterogeneous current of electric through too much synchronous rotating angle, map and involve each harmonic synchronous rotating frame to corresponding mutually orthogonal base, DAZ gene is realized respectively, and superposition forms complete current controller by proportional and integral controller. The method advantage is: adopts trapezoidal integration to improve integral accuracy, and is aided with variable-structure control, and introducing suitable self-adaptation, to compensate anti-control saturated, further increases dynamic and static state performance; Suppress the loss of electric machine that the harmful subharmonic electric current of heterogeneous motor causes, the negative impacts such as torque pulsation and vibratory noise, and the positive effect making available subharmonic is excavated; Give full play to the advantage of heterogeneous motor, it is to increase system performance.
Description
Technical field
The invention belongs to alternating current motor and control techniques field thereof, it is specifically related to a kind of multiphase induction motor and specifies subharmonic electric current suppressing method, to realize current waveform optimal control, can be applicable to the high-power AC motor transmission occasions such as ship propulsion, mining machinery, track drawing, steel rolling, pump and fan speed governing.
Background technology
The alternating current machine feature that especially cage-type induction machine simply, is easily safeguarded with its structure and cost is cheap is widely used. And the waste day by day highlighting the high-power AC motor making employing speed governing transmission reduction account for very big proportion in power consumption of energy problem and loss are to realize energy-saving and emission-reduction, there is great economy and social effect, day by day paid close attention to.
Along with the development of Power Electronic Technique, multiphase power converter makes motor break away from the restriction of the industrial electrical network number of phases. Compared with conventional three-phase electric system, the significant advantage of multiphase system is:
Along with the raising of power grade, heterogeneous motor is by increasing number of motor phases and inverter leg number, the high-power operation of system is not only realized easily through low-voltage device, be conducive to the cost of attenuating system, also avoid the problems such as quiet dynamic voltage balancing problem that devices in series brings and the topology that many level bring and control thereof are complicated. Especially in the occasion (such as marine electric power propulsion) that service voltage is limited, advantage is had more.
Heterogeneous motor has phase redundancy characteristic, when phase fault, still can obtain stable torque, it is achieved fault-tolerant anti-interference operation. It is suitable for naval vessel to advance. Nuclear power plants water-cooling system, the occasion of the high reliability request such as aerospace.
Heterogeneous motor has and more controls resource. From switch state, there is more non-zero voltage space vector, from coordinate transform, there is multiple mutually orthogonal dq plane, wherein can control corresponding variable respectively, have more potential.
In heterogeneous motor, harmonic field number of times increases, and impact reduces, and with the torque pulsation that galvanic action produces, also reduces so that vibration and noise reduces, and rotor loss is lowered.
But, in general the heterogeneous motor number of turn that is often in series is less, and stator leakage inductance is relatively little, to the restraining effect of current harmonics under identical circumstances not as good as three-phase machine.
Current harmonics mainly comes from: the voltage source converter non-linear character of power supply and the dead time effect etc. of PWM can make output voltage contain a large amount of low-order harmonics, harmonic voltage acts on low resistance spatially can produce very big harmonic current, in addition, there is intrinsic non-linear character in the iron core of motor, the harmonic wave counter potential that non-sinusoidal air gap magnetic flux causes all produces harmonic current in motor runs, and, winding etc. opened by some heterogeneous motor New Topological such as H bridges, owing to having untied winding neutral point, some times harmonic electric current is made to form path, it has been impelled to be formed.
Multiphase induction motor is by rational design of Windings, mate its magnetic potential spatial wave and current time ripple to utilize some low-order harmonic to realize energy converting between mechanical, optimize air gap magnetic density waveform, it is to increase utilization ratio unshakable in one's determination, and increase torque density, but the impact of its harmful subharmonic still should not be underestimated. Main reflection exists: increase winding loss; Cause the tangential torque pulsation of motor; Produce the radial electro-magnetic exciting force of motor, increase vibratory noise; Cause phase current peak value to increase, and over current fault may be caused, reduce electric system performance. Although employing wave filter can address this problem to a certain extent, but result in the increase of system bulk, weight, cost undoubtedly.
The control strategy of multiphase induction motor mainly comprises constant voltage constant frequency control, field-oriented vector control, Direct Torque Control etc. Wherein magnetic linkage and electromagnetism torque are adopted closed-loop control by coordinate transform by vector controlled respectively, it is achieved the decoupling zero of magnetic field and electric current, have good dynamic and static state performance; Direct Torque Control does not need complicated coordinate transform, by the direct tracking of magnetic linkage and torque, it is achieved the high dynamic performance of PWM and system, has robustness.
Keeping in check in the disadvantageous effect of the harmful subharmonic electric current of multiphase induction motor, its application is restricted. And at present, rarely have corresponding technical literature report about the control strategy solving multiphase induction motor appointment subharmonic electric current suppression.
Summary of the invention
It is an object of the invention to propose a kind of multiphase induction motor and specify subharmonic electric current suppressing method, to realize current waveform optimal control, reduce winding loss; Slacken motor torque ripple; Reduce the radial electro-magnetic exciting force of motor, reduce vibratory noise; The over current fault that the phase current peak value avoiding current harmonics to cause increases and cause, it is to increase electric system performance.
The technical scheme of the present invention is:
A kind of multiphase induction motor specifies subharmonic electric current suppressing method, comprises the following steps:
A, current reference value process: based on multiphase induction motor vector controlled, according to motor rotational speed omegarWith rotor flux ��rProportional integral regulates result, it is determined that base involves the dq component reference value of available subharmonic electric current at respective synchronous rotating frameAnd make harmful subharmonic electric current of specifying in the dq component reference value of respective synchronous rotating frameIt is zero; Described available subharmonic electric current refers to the low-order harmonic electric current that can realize energy converting between mechanical, harmful subharmonic electric current refers to the higher harmonic electric current that can not realize energy converting between mechanical, l is available harmonic order, h is unwanted harmonics number of times, 1��l < h, the minimum number of times of unwanted harmonics is higher than the available the highest number of times of harmonic wave;
B, current feedback value process: by each for motor phase current feedback value ia��imThrough heterogeneous many synchronous rotating angle, map and involve each harmonic synchronous rotating frame to mutually orthogonal base, obtain base and involve each harmonic electric current dq component value of feedback idl��iql��idh��iqh;
C, current setting: the electric current dq component value of feedback that the electric current dq component reference value determined according to steps A and step B determine regulates in the proportional integral that corresponding synchronous rotating frame carries out improving separately, it is realized by antisaturation proportional and integral controller that the proportional integral improved regulates, multiple independent synchronous rotating frame current regulator superposition is formed complete current controller, regulates and produce each synchronous rotating frame voltage dq component reference value
D, voltage reference value are determined: each synchronous rotating frame voltage dq component reference value determined according to step C, through heterogeneous many synchronously rotating reference frames inverse transformation, forms each cross streams voltage reference value
Further technical scheme is:
Described multiphase induction motor specifies subharmonic electric current suppressing method, and the heterogeneous many synchronous rotating angle of its step B realize coordinate transform by matrix T:
[idliql��idniqn]T=T(n+1)��m[iaib��im]T
Matrix T is following form:
Wherein, n is the highest number of times of unwanted harmonics specified, and m is the number of phases, ��=2 ��/m, is each alternate angle, ��1For the phasing degree of base wave rotor flux linkage vector and static coordinate system a phase axis, ��3=3 ��1, ��5=5 ��1...=..., ��n=n ��1��
Described multiphase induction motor specifies subharmonic electric current suppressing method, and the heterogeneous many synchronously rotating reference frames inverse transformation of its step D is by the inverse matrix T of matrix T-1Realize coordinate transform:
Described multiphase induction motor specifies subharmonic electric current suppressing method, and its step C ratio integral regulator adopts trapezoidal integration, introduces structure changes and compensates, it is achieved the self-adaptative adjustment of integral, and anti-control is saturated, has following transport function:
Wherein, kpFor scale-up factor, kiFor integral coefficient, e is setter bias term, unFor the unsaturated output item of setter, usFor the Saturated output item of setter, saturated bias term is un-us=-es, coefficient �� realizes the self-adaptative adjustment of integral,The adaptive change rule of �� is:
Factor alpha > 0 in formula.
The remarkable technique effect of the present invention is described as follows by technical scheme and principle in conjunction with the present invention:
A kind of multiphase induction motor of the present invention specify subharmonic electric current suppressing method know-why mainly: the present invention is based on the rotor flux linkage orientation vector controlled of the many synchronous rotating frames of multiphase induction motor, utilize the multi-control degree of freedom of heterogeneous motor, by rotary current through heterogeneous many synchronous rotating angle, map and involve each harmonic synchronous rotating frame to mutually orthogonal base, its dq component is all characterized by the DC quantity of decoupling zero, it is easy to adopt structure simple, parameter designing is easy, the proportional and integral controller of strong robustness realizes quick DAZ gene respectively, and simple superposition forms complete current controller, tradition proportional and integral controller relatively, trapezoidal integration discretize is adopted to realize in digital display circuit, improve control accuracy, and adopt variable-structure control, introduce the suitable anti-control of self-adaptation compensation tache saturated, further increase dynamic and static state performance. it is simple and practical that the present invention controls method, it is not necessary to complicated design of Regulator can realize uneoupled control, and can not cause the mutual interference between different frequency harmonics and cause system unstable. the loss of electric machine that the harmful subharmonic electric current of this control method suppression multiphase induction motor causes, the negative impacts such as torque pulsation and vibratory noise, and make the positive effect of available subharmonic (as improved torque density, optimize magnetic close to improve iron core utilization etc.) excavated, give full play to the advantage of heterogeneous motor, it is to increase system performance.
Accompanying drawing explanation
Fig. 1 is that described multiphase induction motor specifies subharmonic electric current suppressing method always to scheme.
Fig. 2 is that described multiphase induction motor specifies subharmonic electric current antisaturation proportional integral under many synchronous rotating frames to regulate suppressing method schematic diagram.
Embodiment
It is as follows that the invention will be further described in conjunction with the accompanying drawings and embodiments:
Embodiment 1: as shown in Figure 1, 2 is a basic embodiment of the present invention. A kind of multiphase induction motor specifies subharmonic electric current suppressing method, comprises the following steps:
A, current reference value process: based on the rotor flux linkage orientation vector controlled of the many synchronous rotating frames of multiphase induction motor, according to motor rotational speed omegarWith rotor flux ��rProportional integral regulates result, determine that motor base involves ratio and the phase relationship of available subharmonic electric current, extract, at respective synchronous rotating frame, the magnetic flux density waveforms that current excitation component makes it to produce expectation, obtain base and involve the dq component reference value of available subharmonic electric current at respective synchronous rotating frameAnd make harmful subharmonic electric current of specifying in the dq component reference value of respective synchronous rotating frameIt is zero; Described available subharmonic electric current refers to can realize energy converting between mechanical, optimize air gap magnetic density waveform, increase torque density, improve the low-order harmonic electric current of utilization ratio unshakable in one's determination, harmful subharmonic electric current refers to can not realize energy converting between mechanical, increase winding loss, cause tangential torque pulsation, produce radial electro-magnetic exciting force, increase the higher harmonic electric current of vibratory noise, l is available harmonic order, during l=1, it is base ripple, h is unwanted harmonics number of times, 1��l < h, the minimum number of times of unwanted harmonics is higher than the available the highest number of times of harmonic wave;
B, current feedback value process: by each for motor phase current feedback value ia��im, wherein involve available subharmonic containing base, and the harmful subharmonic of superposition, through heterogeneous many synchronous rotating angle, map and involve each harmonic synchronous rotating frame to mutually orthogonal base, obtain base and involve each harmonic electric current dq component value of feedback idl��iql��idh��iqh, all it is characterized by DC quantity;
C, current setting: the electric current dq component reference value determined according to steps AWith the electric current dq component value of feedback i that step B determinesdl��iql��idh��iqhIt is the DC quantity of decoupling zero at corresponding synchronous rotating frame separately, the proportional integral carrying out improving regulates, realize the quick non-corresponding control that base involves each harmonic electric current dq component, it is realized by antisaturation proportional and integral controller that the proportional integral improved regulates, multiple independent synchronous rotating frame current regulator superposition is formed complete current controller, regulates and produce each synchronous rotating frame voltage dq component reference value
D, voltage reference value are determined: each synchronous rotating frame voltage dq component reference value determined according to step CThrough heterogeneous many synchronously rotating reference frames inverse transformation, form each cross streams voltage reference valueCarry out Realization of pulse width modulation based on voltage space vector (SVPWM), drive multi-phase voltage source invertor, power to multiphase induction motor.
Embodiment 2: be the further embodiment of embodiment 1. As different from Example 1: the heterogeneous many synchronous rotating angle of described step B realize heterogeneous static coordinate by matrix T to be tied to the coordinate transform of many synchronous rotating frames:
[idliql��idniqn]T=T(n+1)��m[iaib��im]T
Matrix T is following form:
Wherein, n is the highest number of times of unwanted harmonics specified, and m is the number of phases, ��=2 ��/m, is each alternate angle, ��1For the phasing degree of base wave rotor flux linkage vector and static coordinate system a phase axis, ��3=3 ��1, ��5=5 ��1...=..., ��n=n ��1��
The heterogeneous many synchronously rotating reference frames inverse transformation of described step D is by the inverse matrix T of matrix T-1Realize many synchronous rotating frames to the coordinate transform of heterogeneous static coordinate system:
Described step C ratio integration (PI) setter has following transport function:
Wherein, kpFor scale-up factor, kiFor integral coefficient, utilizing the method for pole zero cancellation to be designed by regulator parameter, when digital control system realizes, adopt trapezoidal integration to improve the precision of integral, eliminate quiet difference, its increment type pi regulator discrete form is:
Wherein, k is sampling sequence number, and e is setter bias term, and u is setter output item, and �� is for exporting increment; And introduce structure changes and compensate, it is achieved the self-adaptative adjustment of integral, anti-control is saturated, and transport function is:
unFor the unsaturated output item of setter, usFor the Saturated output item of setter, saturated bias term is un-us=-es, coefficient �� realizes the self-adaptative adjustment of integral,The adaptive change rule of �� is:
Factor alpha > 0 in formula.
Embodiment 3: be the present invention's preferred embodiment. Multiphase induction motor is that five phase sines drive winding power induction motor, and fundamental current realizes energy converting between mechanical, and triple harmonic current does not produce energy converting between mechanical, for suppressing object. The multiphase induction motor of the present embodiment specifies subharmonic electric current suppressing method to comprise the following steps:
A, current reference value process: based on the rotor flux linkage orientation vector controlled of five phase induction motor base ripple synchronous rotating frames, according to motor rotational speed omegarWith rotor flux ��rProportional integral regulates result, it is determined that motor base ripple is in the dq component reference value of its synchronous rotating frameAnd make the triple harmonic current specified in the dq component reference value of its synchronous rotating frameIt is zero, is eliminated;
B, current feedback value process: by motor five phase current feedback value ia��is, wherein containing base ripple, and superposition triple-frequency harmonics, through five mutually more synchronous rotating angle, mapping mutually orthogonal base from five phase static coordinate systems and involve triple-frequency harmonics synchronous rotating frame, conversion realizes by matrix T:
[id1iq1��id3iq3]T=T4��5[iaib��is]T
T is following form:
Wherein, ��1For the phasing degree of base wave rotor flux linkage vector and static coordinate system a phase axis;
Conversion obtains base and involves triple harmonic current dq component value of feedback id1��iq1��id3��iq3, all it is characterized by DC quantity;
C, current setting: the electric current dq component reference value determined according to steps AWith the electric current dq component value of feedback i that step B determinesd1��iq1��id3��iq3The DC quantity being decoupling zero, carries out proportional integral adjustment, it is achieved base involves the quick non-corresponding control of triple harmonic current dq component, it is that passing ratio integral regulator realizes that proportional integral regulates, and proportional integral (PI) setter has following transport function:
Wherein, kpFor scale-up factor, kiFor integral coefficient, utilizing the method for pole zero cancellation to be designed by regulator parameter, when digital control system realizes, adopt trapezoidal integration to improve the precision of integral, eliminate quiet difference, its increment type pi regulator discrete form is:
Wherein, k is sampling sequence number, and e is setter bias term, and u is setter output item, and �� is for exporting increment; Introducing structure changes compensates, it is achieved the self-adaptative adjustment of integral, and anti-control is saturated, and transport function is:
unFor the unsaturated output item of setter, usFor the Saturated output item of setter, saturated bias term is un-us=-es, coefficient �� realizes the self-adaptative adjustment of integral,The adaptive change rule of �� is:
Factor alpha > 0 in formula.
Ji Bo and triple-frequency harmonics synchronous rotating frame current regulator superposition are formed complete current controller, regulates and produce base ripple and triple-frequency harmonics synchronous rotating frame voltage dq component reference value
D, voltage reference value are determined: the base ripple determined according to step C and triple-frequency harmonics synchronous rotating frame voltage dq component reference valueThrough five mutually more synchronously rotating reference frame inverse transformations, it is achieved Ji Bo and triple-frequency harmonics synchronous rotating frame are to the conversion of five phase static coordinate systems, and conversion is by the inverse matrix T of matrix T-1Realize:
T-1For following form:
Wherein, ��1For the phasing degree of base wave rotor flux linkage vector and static coordinate system a phase axis;
Conversion obtains five cross streams voltage reference valuesCarry out Realization of pulse width modulation based on voltage space vector (SVPWM), drive multi-phase voltage source invertor, power to multiphase induction motor.
Embodiment 4: be another preferred embodiment of the present invention. Multiphase induction motor is that five phase concentrated and full-pitch windings star types connect non-sinusoidal power supply induction motor, fundamental current realizes energy converting between mechanical, triple harmonic current can be utilized to realize energy converting between mechanical simultaneously, optimizes air gap magnetic density waveform, increase torque density, it is to increase utilization ratio unshakable in one's determination; Five subharmonic electric currents connect due to winding star type, can not form path and cannot circulate, be zero, do not control, and the seventh harmonic electric current does not realize energy converting between mechanical, will produce torque pulsation, loss etc., for suppressing object. The multiphase induction motor of the present embodiment specifies subharmonic electric current suppressing method to comprise the following steps:
A, current reference value process: the rotor flux linkage orientation vector controlled involving triple-frequency harmonics synchronous rotating frame based on five phase induction motor bases, according to motor rotational speed omegarWith rotor flux ��rProportional integral regulates result, determine that base involves ratio and the phase relationship of triple harmonic current, involve triple-frequency harmonics synchronous rotating frame at base and extract the magnetic flux density waveforms that current excitation component makes it to produce expectation, obtain the dq component reference value that base involves triple-frequency harmonics synchronous rotating frameAnd make the seventh harmonic electric current specified in the seventh harmonic synchronous rotating frame dq component reference valueIt is zero, is eliminated;
B, current feedback value process: by motor five phase current feedback value ia��is, wherein containing base ripple, and superposition three, the seventh harmonic, through five mutually more synchronous rotating angle, from five phase static coordinate systems map mutually orthogonal base involve three, five, the seventh harmonic synchronous rotating frame, conversion realizes by matrix T:
[id1iq1��id7iq7]T=T8��5[iaib��is]T
T is following form:
Wherein, ��1For the phasing degree of base wave rotor flux linkage vector and static coordinate system a phase axis;
Conversion obtains that base involves three, the seventh harmonic electric current dq component value of feedback id1��iq1, id3��iq3, id5��iq5, id7��iq7, all it is characterized by DC quantity, id5��iq5It is zero;
C, current setting: the electric current dq component reference value determined according to steps AWith the electric current dq component value of feedback i that step B determinesd1��iq1, id3��iq3��id7��iq7The DC quantity being decoupling zero, carries out proportional integral adjustment, it is achieved base involves three, the quick non-corresponding control of the seventh harmonic electric current dq component, it is that passing ratio integral regulator realizes that proportional integral regulates, and proportional integral (PI) setter has following transport function:
Wherein, kpFor scale-up factor, kiFor integral coefficient, utilizing the method for pole zero cancellation to be designed by regulator parameter, when digital control system realizes, adopt trapezoidal integration to improve the precision of integral, eliminate quiet difference, its increment type pi regulator discrete form is:
Wherein, k is sampling sequence number, and e is setter bias term, and u is setter output item, and �� is for exporting increment; Introducing structure changes compensates, it is achieved the self-adaptative adjustment of integral, and anti-control is saturated, and transport function is:
unFor the unsaturated output item of setter, usFor the Saturated output item of setter, saturated bias term is un-us=-es, coefficient �� realizes the self-adaptative adjustment of integral,The adaptive change rule of �� is:
Factor alpha > 0 in formula.
Ji Bo and three, the seventh harmonic synchronous rotating frame current regulator superposition are formed complete current controller, regulate produce base ripple and three, the seventh harmonic synchronous rotating frame voltage dq component reference value
D, voltage reference value are determined: the base ripple determined according to step C and three, the seventh harmonic synchronous rotating frame voltage dq component reference valueWith for zero five subharmonic synchronous rotating frame voltage dq component reference valuesThrough five mutually more synchronously rotating reference frame inverse transformations, it is achieved Ji Bo and three, five, the seventh harmonic synchronous rotating frame to the conversion of five phase static coordinate systems, conversion is by the inverse matrix T of matrix T-1Realize:
Conversion obtains five cross streams voltage reference valuesCarry out Realization of pulse width modulation based on voltage space vector (SVPWM), drive multi-phase voltage source invertor, power to multiphase induction motor.
Specific embodiment described herein is only to the present invention's spirit explanation for example. Described specific embodiment can be made various amendment or supplements or adopt similar mode to substitute by those skilled in the art, but can't deviate the spirit of the present invention or surmount described scope as defined in the claims.
Claims (3)
1. a multiphase induction motor specifies subharmonic electric current suppressing method, it is characterised in that, comprise the following steps:
A, current reference value process: based on multiphase induction motor vector controlled, according to motor rotational speed omegarWith rotor flux ��rProportional integral regulates result, it is determined that base involves the dq component reference value of available subharmonic electric current at respective synchronous rotating frameAnd make harmful subharmonic electric current of specifying in the dq component reference value of respective synchronous rotating frameIt is zero; Described available subharmonic electric current refers to the low-order harmonic electric current that can realize energy converting between mechanical, harmful subharmonic electric current refers to the higher harmonic electric current that can not realize energy converting between mechanical, l is available harmonic order, h is unwanted harmonics number of times, 1��l < h, the minimum number of times of unwanted harmonics is higher than the available the highest number of times of harmonic wave;
B, current feedback value process: by each for motor phase current feedback value ia��imThrough heterogeneous many synchronous rotating angle, map and involve each harmonic synchronous rotating frame to mutually orthogonal base, obtain base and involve each harmonic electric current dq component value of feedback idl��iql��idh��iqh;
C, current setting: the electric current dq component value of feedback that the electric current dq component reference value determined according to steps A and step B determine regulates in the proportional integral that corresponding synchronous rotating frame carries out improving separately, it is realized by antisaturation proportional and integral controller that the proportional integral improved regulates, multiple independent synchronous rotating frame current regulator superposition is formed complete current controller, regulates and produce each synchronous rotating frame voltage dq component reference value
D, voltage reference value are determined: each synchronous rotating frame voltage dq component reference value determined according to step C, through heterogeneous many synchronously rotating reference frames inverse transformation, forms each cross streams voltage reference value
Described step C ratio integral regulator adopts trapezoidal integration, introduces structure changes and compensates, it is achieved the self-adaptative adjustment of integral, and anti-control is saturated, has following transport function:
Wherein, kpFor scale-up factor, kiFor integral coefficient, coefficient �� realizes the self-adaptative adjustment of integral,The adaptive change rule of �� is:
In formula, factor alpha > 0, e is setter bias term, unFor the unsaturated output item of setter, usFor the Saturated output item of setter, saturated bias term is un-us=-es.
2. multiphase induction motor according to claim 1 specifies subharmonic electric current suppressing method, it is characterised in that, the heterogeneous many synchronous rotating angle of described step B realize coordinate transform by matrix T:
[id1iq1��idniqn]T=T(n+1)��m[iaib��im]T
Matrix T is following form:
Wherein, n is the highest number of times of unwanted harmonics specified, and m is the number of phases, ��=2 ��/m, is each alternate angle, ��1For the phasing degree of base wave rotor flux linkage vector and static coordinate system a phase axis, ��3=3 ��1, ��5=5 ��1...=..., ��n=n ��1��
3. multiphase induction motor according to claim 2 specifies subharmonic electric current suppressing method, it is characterised in that, the heterogeneous many synchronously rotating reference frames inverse transformation of described step D is by the inverse matrix T of matrix T-1Realize coordinate transform:
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