CN103929109A - High-power built-in permanent magnet synchronous motor position-sensorless control system and control method - Google Patents

Abstract

The invention provides a high-power built-in permanent magnet synchronous motor position-sensorless control system and a control method, and belongs to the field of motor control. The control system solves the problem that when a traditional high-power built-in permanent magnet synchronous motor system works at a low switch frequency, low-order harmonic content increases, and meets the requirements that low-order harmonic waves are weakened and meanwhile, rotating voltage vector control with constant amplitude values is produced. An SHMPWM technique is adopted, it is guaranteed that on a low-switch-frequency work condition, harmonic waves of a high-power permanent magnet synchronous motor system are inhibited, constraint conditions of SHMPWM non-linear transcendental equation sets are increased, and three-phase specific harmonic wave content signals with constant amplitude values and 120-degree phase angle differences are extracted from the weakened harmonic wave content to be used for estimation of a rotor position; specific harmonic currents are obtained through a band-pass filter; by means of a position observer, the estimated rotation speed and the rotor position of a permanent magnet synchronous motor are obtained, and position-sensorless rotation speed closed loop vector control of a motor is achieved. The control system and the control method are suitable for control of the motor.

Description

A kind of high-power internal permanent magnet synchronous motor control system without position sensor and control method

Technical field

The invention belongs to Motor Control Field.

Background technology

The high-power synchronous machine of built-in magnetic forever drive system, because its energy-efficient design concept makes it to have broad application prospects in the high-power electric drive system such as such as civilian use electric vehicle, electric drive armored vehicle, high ferro, is also an important subject in electrical engineering field.

In order to realize the high performance control of permagnetic synchronous motor, must know rotor-position and the velocity information of motor, conventionally obtain by the mechanical sensor such as encoder or resolver.But the installation of mechanical sensor brings the problems such as system cost increase, reliability reduction, size increase and antijamming capability reduction.In order to widen the range of application of permagnetic synchronous motor, the position Sensorless Control that realizes motor is a study hotspot of high-performance motor transmission field always.Wherein, the specific high-frequency signal that utilizes injection or inverter PWM to produce, even also can realize the estimation to rotor information under zero-speed at motor low speed, is the effective means that realizes the control of machinery-free type transducer.

With in, compared with small-power drive system, heavy-duty motor drive system has higher technical requirement aspect a lot.Consider from inverter angle, high-power switch device is subject to the restriction of switching loss and heat radiation, the highest switching frequency is generally limited in hundreds of hertz left and right, but it is even higher that the fundamental frequency of while motor can approach 200Hz sometimes, the reduction of switching frequency can increase the harmonic components of system, the burden of accentuation filter, therefore, under compared with the restriction of low switching frequency, adopt an effective measure to eliminate and be difficult for by the low-order harmonic of filter circuit filtering, the waveform quality of raising system is another study hotspot of heavy-duty motor drive system.

Summary of the invention

The present invention is the problem increasing in order to solve low-order harmonic composition that traditional high-power internal permanent magnet synchronous motor system causes in the time that low switching frequency is worked, also in order to meet the demand of the rotational voltage vector control that produces constant amplitude in the time slackening low-order harmonic simultaneously.A kind of high-power internal permanent magnet synchronous motor control system without position sensor and control method are now provided.

A kind of high-power internal permanent magnet synchronous motor control system without position sensor, this system comprises PI computing unit No. one, for to angular speed with rotor feedback angular velocity omega _rdifference carry out PI computing and obtain given torque

MTPA breakdown torque/ampere unit, for to described given torque carry out the computing of breakdown torque current ratio, and obtain d, q shaft current set-point

No. two PI computing units, for to q shaft current set-point with q shaft current value of feedback i _qdifference carry out PI computing, obtain q shaft voltage set-point

No. three PI computing units, for to d shaft current set-point with d shaft current value of feedback i _ddifference carry out PI computing, obtain d shaft voltage set-point

A Park inverse transformation block, for the positional information to rotor described q shaft voltage set-point with d shaft voltage set-point carry out Park inverse transformation, obtain respectively the given voltage of α axle with the given voltage of β axle

Three level SHMPWM unit, for to the given voltage of described α axle with the given voltage of β axle carry out three level SHMPWM computings, and then the pulse width signal of generating power device;

Three-level inverter unit, for generating three-phase tri-level voltage signal according to described power device pulse width signal, and exports the current i of a phase and b phase _a, i _b, and in described three-phase tri-level signal, comprise the m subharmonic voltage vector u at constant amplitude, 120 ° of phase angles of phase phasic difference _am, u _bm, u _cm;

A Clark converter unit and No. two Park inverse transformation block, for by the positional information of rotor described three-phase current signal i _a, i _bcarry out successively Clark conversion and Park inverse transformation, obtain d shaft current value of feedback i _dwith q shaft current value of feedback i _q;

Band pass filter BPF, for to described three-phase current signal i _a, i _bfiltering, obtains particular harmonic current i _am, i _bm;

No. two Clark converter units, for to described particular harmonic current i _am, i _bmcarry out Clark conversion, obtain the current i under alpha-beta axle system _{α m}, i _{β m};

Position detection device unit, for the current i under to described alpha-beta axle being _{α m}, i _{β m}carry out signal processing, obtain rotor feedback angular velocity omega _rand the positional information of rotor

A kind of high-power internal permanent magnet synchronous motor method for controlling position-less sensor, the method comprises the following steps:

Step 1, angular speed with rotor feedback angular velocity omega _rit is poor to do, and this difference is carried out to PI computing and obtain given torque

Step 2, to described given torque carry out the computing of MTPA breakdown torque current ratio, obtain d, q shaft current set-point

Step 3, to q shaft current set-point with q shaft current value of feedback i _qdifference carry out PI computing, obtain q shaft voltage set-point to d shaft current set-point with d shaft current value of feedback i _ddifference carry out PI computing, obtain d shaft voltage set-point

Step 4, to described q shaft voltage set-point d shaft voltage set-point positional information with rotor carry out Park inverse transformation, obtain respectively the given voltage of α axle with the given voltage of β axle

Step 5, according to the given voltage of described α axle with the given voltage of β axle obtain modulation ratio, carry out three level SHMPWM computings, and then the pulse width signal of generating power device;

Step 6, export the m subharmonic voltage vector u that contains constant amplitude, 120 ° of phase angles of phase phasic difference by three-level inverter unit according to described power device pulse width signal _am, u _bm, u _cmthree-phase tri-level voltage signal, this three-phase tri-level voltage signal is added on three phase windings of high-power PMSM, realizes the vector control of PMSM, on motor winding, produces three-phase current signal, and exports the current i of a phase and b phase _a, i _b;

Step 7, current i to a phase described in step 6 and b phase _a, i _bcarry out successively Clark conversion and Park inverse transformation, obtain d shaft current value of feedback i _dwith q shaft current value of feedback i _q;

Step 8, the current i of employing band pass filter BPF to described a phase and b phase _a, i _bfiltering, obtains particular harmonic current i _am, i _bm;

Step 9, to described particular harmonic current i _am, i _bmcarry out Clark conversion, obtain the current i under alpha-beta axle system _{α m}, i _{β m};

Step 10, position detection device unit are to the current i under described alpha-beta axle system _{α m}, i _{β m}carry out signal processing, obtain rotor feedback angular velocity omega _rand the positional information of rotor complete once high-power internal permanent magnet synchronous motor position Sensorless Control, and return to execution step one, carry out high-power internal permanent magnet synchronous motor position Sensorless Control next time.

The high-power internal permanent magnet synchronous motor control system without position sensor of one of the present invention and control method, under traditional SHEPWM modulator approach, increase constraints, even if high-power internal permanent magnet synchronous motor system works is at low switching frequency, the problem that the low-order harmonic composition still can filtering being caused by low switching frequency increases.

Adopt particular harmonic to cut down pulse-width modulation (SHMPWM) technology, ensure under low switching frequency condition of work, the harmonic wave of high power permanent magnet synchronous motor system to be suppressed; And by strengthening the constraints of SHMPWM non-linear transcendental equation group, the three-phase particular harmonic of extracting 120 ° of constant amplitude, phase angle mutual deviation from the harmonic components weakening becomes sub-signal for carrying out the estimation of rotor-position; Adopt band pass filter (BPF) to obtain particular harmonic and become the current response i of sub-signal _{α c}, i _{β c}; Utilize position detection device to obtain permagnetic synchronous motor rotating speed and the rotor-position estimated, realize the position-sensor-free speed closed loop vector control of motor.

Brief description of the drawings

Fig. 1 is control system system block diagram of the present invention;

Fig. 2 is three-level inverter main circuit schematic diagram;

Fig. 3 is the switching mode figure of three level SHMPWM;

Fig. 4 is internal permanent magnet synchronous motor phasor coordinate figure.

Embodiment

Embodiment one, seeing figures.1.and.2 illustrates present embodiment, the high-power internal permanent magnet synchronous motor control system without position sensor of one described in present embodiment, and this system comprises PI computing unit 1 No. one, for to angular speed with rotor feedback angular velocity omega _rdifference carry out PI computing and obtain given torque

MTPA breakdown torque/ampere unit 2, for to described given torque carry out the computing of breakdown torque current ratio, and obtain d, q shaft current set-point

No. two PI computing units 3, for to q shaft current set-point with q shaft current value of feedback i _qdifference carry out PI computing, obtain q shaft voltage set-point

No. three PI computing units 4, for to d shaft current set-point with d shaft current value of feedback i _ddifference carry out PI computing, obtain d shaft voltage set-point

A Park inverse transformation block 5, for the positional information to rotor described q shaft voltage set-point with d shaft voltage set-point carry out Park inverse transformation, obtain respectively the given voltage of α axle with the given voltage of β axle

Three level SHMPWM unit 6, for to the given voltage of described α axle with the given voltage of β axle carry out three level SHMPWM computings, and then the pulse width signal of generating power device;

Three-level inverter unit 7, for generating three-phase tri-level voltage signal according to described power device pulse width signal, and exports the current i of a phase and b phase _a, i _b, and in described three-phase tri-level signal, comprise the m subharmonic voltage vector u at constant amplitude, 120 ° of phase angles of phase phasic difference _am, u _bm, u _cm;

A Clark converter unit 8 and No. two Park inverse transformation block 9, for by the positional information of rotor described three-phase current signal i _a, i _bcarry out successively Clark conversion and Park inverse transformation, obtain d shaft current value of feedback i _dwith q shaft current value of feedback i _q;

Band pass filter BPF10, for to described three-phase current signal i _a, i _bfiltering, obtains particular harmonic current i _am, i _bm; No. two Clark converter units 11, for to described particular harmonic current i _am, i _bmcarry out Clark conversion, obtain the current i under alpha-beta axle system _{α m}, i _{β m};

Position detection device unit 12, for the current i under to described alpha-beta axle being _{α m}, i _{β m}carry out signal processing, obtain rotor feedback angular velocity omega _rand the positional information of rotor

The method for controlling position-less sensor of a kind of high-power internal permanent magnet synchronous motor control system without position sensor described in embodiment two, this embodiment, the method comprises the following steps:

Step 1, angular speed with rotor feedback angular velocity omega _rit is poor to do, and this difference is carried out to PI computing and obtain given torque

Step 2, to described given torque carry out the computing of MTPA breakdown torque current ratio, obtain d, q shaft current set-point

Step 3, to q shaft current set-point with q shaft current value of feedback i _qdifference carry out PI computing, obtain q shaft voltage set-point to d shaft current set-point with d shaft current value of feedback i _ddifference carry out PI computing, obtain d shaft voltage set-point

Step 4, to described q shaft voltage set-point d shaft voltage set-point positional information with rotor carry out Park inverse transformation, obtain respectively the given voltage of α axle with the given voltage of β axle

Step 5, according to the given voltage of described α axle with the given voltage of β axle obtain modulation ratio, carry out three level SHMPWM computings, and then the pulse width signal of generating power device;

Step 6, export the m subharmonic voltage vector u that contains constant amplitude, 120 ° of phase angles of phase phasic difference by three-level inverter unit according to described power device pulse width signal _am, u _bm, u _cmthree-phase tri-level voltage signal, this three-phase tri-level voltage signal is added on three phase windings of high-power PMSM, realizes the vector control of PMSM, on motor winding, produces three-phase current signal, and exports the current i of a phase and b phase _a, i _b;

Step 7, current i to a phase described in step 6 and b phase _a, i _bcarry out successively Clark conversion and Park inverse transformation, obtain d shaft current value of feedback i _dwith q shaft current value of feedback i _q;

Step 8, the current i of employing band pass filter BPF to described a phase and b phase _a, i _bfiltering, obtains particular harmonic current i _am, i _bm;

Step 9, to described particular harmonic current i _am, i _bmcarry out Clark conversion, obtain the current i under alpha-beta axle system _{α m}, i _{β m};

Step 10, position detection device unit are to the current i under described alpha-beta axle system _{α m}, i _{β m}carry out signal processing, obtain rotor feedback angular velocity omega _rand the positional information of rotor complete once high-power internal permanent magnet synchronous motor position Sensorless Control, and return to execution step one, carry out high-power internal permanent magnet synchronous motor position Sensorless Control next time.

Embodiment three, illustrate present embodiment with reference to Fig. 3 and Fig. 4, present embodiment is further illustrating a kind of high-power internal permanent magnet synchronous motor method for controlling position-less sensor described in embodiment two, in present embodiment, described in step 5 according to the given voltage of described α axle with the given voltage of β axle obtain modulation ratio, carry out three level SHMPWM computings, and then export the m subharmonic voltage vector u that contains constant amplitude, 120 ° of phase angles of phase phasic difference by three-level inverter unit according to described power device pulse width signal described in the process of the pulse width signal of generating power device and step 6 _am, u _bm, u _cmthe acquisition process of three-phase tri-level voltage signal as follows:

Step one by one, be located at and in [0, pi/2] the individual cycle, have N half-convergency, a _i(i=1,2,3 ..., N) and be half-convergency, decompose the amplitude H of inverter output voltage base component of degree n n according to Fourier series ₁and the amplitude H of j subharmonic composition _jwith half-convergency a _ibetween relational expression be

$H_1=\frac{4}{\mathrm{\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^i\sin \left({\mathrm{ja}}_i\right)\right]---\left(2\right)$

$H_j=\frac{4}{\mathrm{j\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^i\sin \left({\mathrm{ja}}_i\right)\right]---\left(3\right);$

Step 1 two, set up the non-linear transcendental equation group of SHMPWM modulation strategy,

|M _a-H ₁|≤L ₁

$\frac{1}{\left|H_1\right|}\frac{4}{\mathrm{j\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^i\sin \left({\mathrm{ja}}_i\right)\right]\≤L_j$ $\frac{1}{\left|H_1\right|}\frac{4}{\mathrm{m\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^i\sin \left({\mathrm{ja}}_i\right)\right]{=L}_m---\left(4\right);$

Wherein M _afor modulation ratio; J=5,7,9 ..., m-2, m+2 ...; M is the number of times of realizing the required particular harmonic of position Sensorless Control and become sub-signal, and m is odd number; L _j, L _mthe harmonic content value allowing under IEEE519-1992 standard for N-1 kind particular harmonic composition;

Step 1 three, by the solving of the non-linear transcendental equation group in step 1 two, solution procedure adopts the discrete method solving of optimization, obtains different modulating and compares M _athe position of the switching angle of the N in situation, and make the discrete question blank of SHMPWM algorithm;

Step 1 four, utilize the given voltage of α axle with the given voltage of β axle calculate the size of required modulation ratio, the switching angle position of SHMPWM algorithm while searching this state in the discrete question blank in step 1 three, described switching angle position is the pulse width signal of power device; Produce three level voltage signals according to this three-level inverter unit, switching angle position, in this voltage signal, comprise the m subharmonic voltage vector u of constant amplitude _am, u _bm, u _cm, N-1 kind harmonic wave has disappeared simultaneously.

The course of work described in present embodiment, has described the process of the m subharmonic voltage vector that obtains constant amplitude in detail, and the m subharmonic voltage vector of this constant amplitude is for follow-up rotor position information calculating.

Embodiment four, present embodiment are further illustrating a kind of high-power internal permanent magnet synchronous motor method for controlling position-less sensor described in embodiment two, in present embodiment, in step 10, position observer unit is to the current i under described alpha-beta axle system _{α m}, i _{β m}carry out signal processing, obtain rotor feedback angular velocity omega _rand the positional information of rotor concrete steps comprise:

Steps A, the particular harmonic composition current model of structure internal permanent magnet synchronous motor under alpha-beta axle system;

Step B, in internal permanent magnet synchronous motor model, introduce the static reference frame γ-δ of two-phase axle system, obtain the particular harmonic composition current model under γ-δ axle system;

Wherein, γ-δ axle system closes with the coordinate of alpha-beta axle system and is: γ axle is ahead of 45 °, α axle, and δ axle is ahead of 90 °, γ axle;

Step C, structure three-phase m subharmonic voltage equation, and select m subharmonic voltage as particular harmonic voltage;

Step D, utilize principle of coordinate transformation and the cosine law, obtain the m subharmonic composition current envelope curve equation under alpha-beta and γ-δ axle are according to the particular harmonic composition current model under the lower particular harmonic composition current model of the alpha-beta axle system in steps A, γ-δ axle system in step B and the three-phase m subharmonic voltage equation in step C;

Step e, go out rotor-position according to described m subharmonic composition current envelope curve equation inference under alpha-beta and γ-δ axle system estimation equation;

Step F, to described rotor-position carry out differential, obtain rotor feedback angular velocity omega _r;

Step G, general and ω _rfeed back to control system, realize the control to position-sensor-free motor speed.

Embodiment five, present embodiment are further illustrating a kind of high-power internal permanent magnet synchronous motor method for controlling position-less sensor described in embodiment four, in present embodiment, the particular harmonic composition current model under alpha-beta axle system in steps A is:

$\left[\begin{array}{c}{i}_{\mathrm{\αm}}\\ i_{\mathrm{\βm}}\end{array}\right]=\left[\begin{array}{cc}{L}_0-L_1\cos \left(2{\mathrm{\θ}}_e\right)& -L_1\sin \left({2\mathrm{\θ}}_e\right)\\ -L_1\sin \left({2\mathrm{\θ}}_e\right)& L_0+L_1\cos \left({2\mathrm{\θ}}_e\right)\end{array}\right]\·\frac{1}{L_0^2-L_1^2}\·\left[\begin{array}{c}{\∫u}_{\mathrm{\αm}}\mathrm{dt}\\ \∫u_{\mathrm{\βm}}\mathrm{dt}\end{array}\right];$

Wherein, c represents radio-frequency component component; L ₀for average inductance, L ₀=(L _d+ L _q)/2; L ₁be half poor inductance, L ₁=(L _d-L _q)/2; L _d, L _qbe respectively d axle inductive component and the q axle inductive component of motor; θ _efor the space potential angle setting between stator a phase axis and rotor d axle.

Embodiment six, present embodiment are further illustrating a kind of high-power internal permanent magnet synchronous motor method for controlling position-less sensor described in embodiment four, in present embodiment, the particular harmonic composition current model under the γ-δ axle system in step B is:

$\left[\begin{array}{c}{i}_{\mathrm{\γm}}\\ i_{\mathrm{\δm}}\end{array}\right]=\left[\begin{array}{cc}{L}_0-L_1\sin \left(2{\mathrm{\θ}}_e\right)& L_1\cos \left({2\mathrm{\θ}}_e\right)\\ L_1\cos \left({2\mathrm{\θ}}_e\right)& L_0+L_1\sin \left({2\mathrm{\θ}}_e\right)\end{array}\right]\·\frac{1}{L_0^2-L_1^2}\·\left[\begin{array}{c}{\∫u}_{\mathrm{\γm}}\mathrm{dt}\\ \∫u_{\mathrm{\δm}}\mathrm{dt}\end{array}\right]---\left(8\right).$

Embodiment seven, present embodiment are further illustrating a kind of high-power internal permanent magnet synchronous motor method for controlling position-less sensor described in embodiment four, in present embodiment, in step C, build m subharmonic voltage equation, and select m subharmonic voltage as follows as the selection foundation of particular harmonic voltage:

According to Fourier analysis, the voltage expression formula of the three-phase m subharmonic voltage composition of three-level inverter output is:

u _am＝H _msin(mωt)

u _bm＝H _msin(mωt-m×120°)

u _cm＝H _msin(mωt+m×120°) (5)

Wherein, H _mfor m subharmonic becomes the amplitude of component voltage, H _mwith M _a, L _mand three-level inverter DC bus-bar voltage U _dcrelevant, H _m=M _al _mu _dc/ 2;

Select m=3k+1, k=1,2,3 ..., formula (5) can be expressed as:

u _am＝H _msin(mωt)

u _bm＝H _msin(mωt-k×360°-120°)＝H _msin(mωt-120°) (6)

u _cm＝H _msin(mωt+k×360°+120°)＝H _msin(mωt+120°)

By formula (5), select m=3k+1, k=1,2,3 ... when subharmonic, the amplitude that can know this m subharmonic is constant, the phase place of the m subharmonic voltage of a phase, b phase and c phase differs the electrical degree of 120 ° in space, as becoming sub-signal for the particular harmonic of rotor position information estimation, and learn thus m ∈ { 7,13,19......}.

Embodiment eight, present embodiment are further illustrating a kind of high-power internal permanent magnet synchronous motor method for controlling position-less sensor described in embodiment four, in present embodiment, the m subharmonic composition current envelope curve equation under alpha-beta and γ-δ axle system in step D is:

M subharmonic composition current envelope curve equation under alpha-beta axle system is:

$\left[\begin{array}{c}{\left|i_{\mathrm{\αm}}\right|}_{\mathrm{peak}}\\ {\left|i_{\mathrm{\βm}}\right|}_{\mathrm{peak}}\end{array}\right]=\frac{3\×H_m}{2\×\mathrm{m\ω}(L_0^2-L_1^2)}\left[\begin{array}{c}\sqrt{L_0^2+L_1^2+{2L}_0{L}_1\cos \left({2\mathrm{\θ}}_e\right)}\\ \sqrt{L_0^2+L_1^2-{2L}_0{L}_1\cos \left({2\mathrm{\θ}}_e\right)}\end{array}\right];$

M subharmonic composition current envelope curve equation under γ-δ axle system is:

$\left[\begin{array}{c}{\left|i_{\mathrm{\γc}}\right|}_{\mathrm{peak}}\\ {\left|i_{\mathrm{\δc}}\right|}_{\mathrm{peak}}\end{array}\right]=\frac{3\×H_m}{2\×\mathrm{m\ω}(L_0^2-L_1^2)}\left[\begin{array}{c}\sqrt{L_0^2+L_1^2+{2L}_0{L}_1\sin \left({2\mathrm{\θ}}_e\right)}\\ \sqrt{L_0^2+L_1^2-{2L}_0{L}_1\sin \left({2\mathrm{\θ}}_e\right)}\end{array}\right].$

Embodiment nine, present embodiment are further illustrating a kind of high-power internal permanent magnet synchronous motor method for controlling position-less sensor described in embodiment four, in present embodiment, in step e, go out rotor-position according to described m subharmonic composition current envelope curve equation inference under alpha-beta and γ-δ axle system the concrete steps of estimation equation comprise:

M subharmonic composition current envelope curve equation under steps A 1, α axle is:

${{\left|i_{\mathrm{\αm}}\right|}_{\mathrm{peak}}}^2=\frac{9\×{H_m}^2}{4\×m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}[L_0^2+L_1^2+{2L}_0{L}_1\cos \left(2{\mathrm{\θ}}_e\right)];$

M subharmonic composition current envelope curve equation under steps A 2, β axle is:

${{\left|i_{\mathrm{\βm}}\right|}_{\mathrm{peak}}}^2=\frac{9\×{H_m}^2}{4\×m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}[L_0^2+L_1^2-{2L}_0{L}_1\cos \left(2{\mathrm{\θ}}_e\right)];$

Steps A 3, by steps A 1 | i _{α m}| _peak ²with steps A 2 | i _{β m}| _peak ²bring following formula into and calculate,

${{\left|i_{\mathrm{\αm}}\right|}_{\mathrm{peak}}}^2-{{\left|i_{\mathrm{\βm}}\right|}_{\mathrm{peak}}}^2=\frac{{{9H}_m}^2{L}_0{L}_1}{m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}\cos \left({2\mathrm{\θ}}_e\right);$

M subharmonic composition current envelope curve equation under steps A 4, γ axle is:

${{\left|i_{\mathrm{\γm}}\right|}_{\mathrm{peak}}}^2=\frac{9\×{H_m}^2}{4\×m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}[L_0^2+L_1^2+{2L}_0{L}_1\sin \left(2{\mathrm{\θ}}_e\right)];$

M subharmonic composition current envelope curve equation under steps A 5, δ axle is:

${{\left|i_{\mathrm{\δm}}\right|}_{\mathrm{peak}}}^2=\frac{9\×{H_m}^2}{4\×m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}[L_0^2+L_1^2-{2L}_0{L}_1\sin \left(2{\mathrm{\θ}}_e\right)];$

Steps A 6, by steps A 4 | i _{γ m}| _peak ²with steps A 5 | i _{δ m}| _peak ²bringing following formula into calculates:

${{\left|i_{\mathrm{\γm}}\right|}_{\mathrm{peak}}}^2-{{\left|i_{\mathrm{\δm}}\right|}_{\mathrm{peak}}}^2=\frac{{{9H}_m}^2{L}_0{L}_1}{m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}\sin \left({2\mathrm{\θ}}_e\right);$

Steps A 7, bring steps A 3 into following formula with the value in steps A 6 and calculate:

$\frac{{{\left|i_{\mathrm{\γm}}\right|}_{\mathrm{peak}}}^2-{{\left|i_{\mathrm{\δm}}\right|}_{\mathrm{peak}}}^2}{{{\left|i_{\mathrm{\αm}}\right|}_{\mathrm{peak}}}^2-{{\left|i_{\mathrm{\βm}}\right|}_{\mathrm{peak}}}^2}=\frac{\sin \left({2\mathrm{\θ}}_e\right)}{\cos \left({2\mathrm{\θ}}_e\right)}=\tan \left({2\mathrm{\θ}}_e\right);$

Steps A 8, can obtain rotor-position by steps A 7 estimation equation be

Embodiment ten: embodiment of present embodiment, present embodiment is introduced a kind of rotor feedback angular velocity omega that obtains in detail _rand the positional information of rotor concrete steps comprise:

α β represents stator rest frame, and dq represents rotor coordinate system, i _a, i _bobtain particular harmonic and become the current response i of sub-signal through band pass filter BPF _ac, i _bc.

1) the particular harmonic Component Model of internal permanent magnet synchronous motor under alpha-beta axle system can be expressed as

$\left[\begin{array}{c}{u}_{\mathrm{\αc}}\\ u_{\mathrm{\βc}}\end{array}\right]=\left[\begin{array}{cc}{L}_0+L_1\cos \left(2{\mathrm{\θ}}_e\right)& L_1\sin \left({2\mathrm{\θ}}_e\right)\\ L_1\sin \left({2\mathrm{\θ}}_e\right)& L_0-L_1\cos \left({2\mathrm{\θ}}_e\right)\end{array}\right]\·\frac{d}{\mathrm{dt}}\·\left[\begin{array}{c}{i}_{\mathrm{\αc}}\\ i_{\mathrm{\βc}}\end{array}\right]---\left(a\right)$

By formula (a), can derive the particular harmonic composition current model under alpha-beta axle system

$\left[\begin{array}{c}{i}_{\mathrm{\αc}}\\ i_{\mathrm{\βc}}\end{array}\right]=\left[\begin{array}{cc}{L}_0-L_1\cos \left(2{\mathrm{\θ}}_e\right)& -L_1\sin \left({2\mathrm{\θ}}_e\right)\\ -L_1\sin \left({2\mathrm{\θ}}_e\right)& L_0+L_1\cos \left({2\mathrm{\θ}}_e\right)\end{array}\right]\·\frac{1}{L_0^2-L_1^2}\·\left[\begin{array}{c}{u}_{\mathrm{\αc}}\mathrm{dt}\\ u_{\mathrm{\βc}}\mathrm{dt}\end{array}\right]---\left(b\right)$

Subscript c represents radio-frequency component component; L ₀for average inductance, L ₀=(L _d+ L _q)/2; L ₁be half poor inductance, L ₁=(L _d-L _q)/2; L _d, L _qbe respectively d axle inductive component and the q axle inductive component of motor; θ e is the space potential angle setting between stator a phase axis and rotor d axle.

2) in order to build position detection device, in motor model, introduce the static reference frame of another two-phase, γ-δ axle system, γ-δ axle system closes with the coordinate of alpha-beta axle system and is: γ axle is ahead of 45 °, α axle, and δ axle is ahead of 90 °, γ axle.The vector correlation of γ-δ axle system as shown in Figure 4.

The particular harmonic Component Model of internal permanent magnet synchronous motor under γ-δ axle system can be expressed as

$\left[\begin{array}{c}{u}_{\mathrm{\γc}}\\ u_{\mathrm{\δc}}\end{array}\right]=\left[\begin{array}{cc}{L}_0+L_1\sin \left(2{\mathrm{\θ}}_e\right)& -L_1\cos \left({2\mathrm{\θ}}_e\right)\\ -L_1\cos \left({2\mathrm{\θ}}_e\right)& L_0-L_1\sin \left({2\mathrm{\θ}}_e\right)\end{array}\right]\·\frac{d}{\mathrm{dt}}\·\left[\begin{array}{c}{i}_{\mathrm{\γc}}\\ i_{\mathrm{\δc}}\end{array}\right]---\left(c\right)$

Formula (c), can derive the particular harmonic composition current model under γ-δ axle system

$\left[\begin{array}{c}{i}_{\mathrm{\γc}}\\ i_{\mathrm{\δc}}\end{array}\right]=\left[\begin{array}{cc}{L}_0-L_1\sin \left(2{\mathrm{\θ}}_e\right)& L_1\cos \left({2\mathrm{\θ}}_e\right)\\ L_1\cos \left({2\mathrm{\θ}}_e\right)& L_0+L_1\sin \left({2\mathrm{\θ}}_e\right)\end{array}\right]\·\frac{1}{L_0^2-L_1^2}\·\left[\begin{array}{c}{\∫u}_{\mathrm{\γc}}\mathrm{dt}\\ \∫u_{\mathrm{\δc}}\mathrm{dt}\end{array}\right]---\left(d\right)$

3) build three-phase m subharmonic voltage equation, and select 7 subharmonic as particular harmonic, the aberration rate L of restriction 7 subharmonic ₇=5%, in the electric system that is 750V at a DC bus, as modulation ratio M _a=1 o'clock, three-phase 7 subharmonic voltage equations were

u _a7＝37.5sin(7ωt)

u _b7＝37.5sin(7ωt-120°) (e)

u _c7＝37.5sin(7ωt+120°)

4) utilize principle of coordinate transformation and the cosine law, by formula (b), (d) and 7 subharmonic composition current envelope curve equations that (e) can be under alpha-beta and γ-δ axle system

$\left[\begin{array}{c}{\left|i_{\mathrm{\αc}}\right|}_{\mathrm{peak}}\\ {\left|i_{\mathrm{\βc}}\right|}_{\mathrm{peak}}\end{array}\right]=\frac{3\×37.5}{2\×7\mathrm{\ω}(L_0^2-L_1^2)}\left[\begin{array}{c}\sqrt{L_0^2+L_1^2+{2L}_0{L}_1\cos \left({2\mathrm{\θ}}_e\right)}\\ \sqrt{L_0^2+L_1^2-{2L}_0{L}_1\cos \left({2\mathrm{\θ}}_e\right)}\end{array}\right]---\left(f\right)$

$\left[\begin{array}{c}{\left|i_{\mathrm{\γc}}\right|}_{\mathrm{peak}}\\ {\left|i_{\mathrm{\δc}}\right|}_{\mathrm{peak}}\end{array}\right]=\frac{3\×37.5}{2\×7\mathrm{\ω}(L_0^2-L_1^2)}\left[\begin{array}{c}\sqrt{L_0^2+L_1^2+{2L}_0{L}_1\sin \left({2\mathrm{\θ}}_e\right)}\\ \sqrt{L_0^2+L_1^2-{2L}_0{L}_1\sin \left({2\mathrm{\θ}}_e\right)}\end{array}\right]---\left(g\right)$

In formula | i _{α c}| _peak, | i _{β c}| _peakfor 7 subharmonic composition current envelope curves of α, β axle;

| i _{γ c}| _peak, | i _{δ c}| _peak---7 subharmonic composition current envelope curves of γ, δ axle;

For internal permanent magnet synchronous motor, because q axle inductance is greater than d axle inductance, i.e. L _d≠ L _q, L ₁≠ 0, cos (2 θ that comprise rotor position information in formula (f), (g) _e) and sin (2 θ _e) Xiang Buwei 0, can derive thus the estimation equation of rotor-position

$\widehat{\mathrm{\θ}}=\frac{1}{2}{\tan }^{-1}\left(\frac{{\left|i_{\mathrm{\γc}}\right|}_{\mathrm{peak}}^2-{\left|i_{\mathrm{\δc}}\right|}_{\mathrm{peak}}^2}{{\left|i_{\mathrm{\αc}}\right|}_{\mathrm{peak}}^2-{\left|i_{\mathrm{\βc}}\right|}_{\mathrm{peak}}^2}\right)---\left(h\right).$

Claims (9)

1. a high-power internal permanent magnet synchronous motor control system without position sensor, is characterized in that, this system comprises a PI computing unit (1), for to angular speed with rotor feedback angular velocity omega _rdifference carry out PI computing and obtain given torque

MTPA breakdown torque/ampere unit (2), for to described given torque carry out the computing of breakdown torque current ratio, and obtain d, q shaft current set-point

No. two PI computing units (3), for to q shaft current set-point with q shaft current value of feedback i _qdifference carry out PI computing, obtain q shaft voltage set-point

No. three PI computing units (4), for to d shaft current set-point with d shaft current value of feedback i _ddifference carry out PI computing, obtain d shaft voltage set-point

A Park inverse transformation block (5), for the positional information to rotor described q shaft voltage set-point with d shaft voltage set-point carry out Park inverse transformation, obtain respectively the given voltage of α axle with the given voltage of β axle

Three level SHMPWM unit (6), for to the given voltage of described α axle with the given voltage of β axle carry out three level SHMPWM computings, and then the pulse width signal of generating power device;

Three-level inverter unit (7), for generating three-phase tri-level voltage signal according to described power device pulse width signal, and exports the current i of a phase and b phase _a, i _b, and in described three-phase tri-level signal, comprise the m subharmonic voltage vector u at constant amplitude, 120 ° of phase angles of phase phasic difference _am, u _bm, u _cm;

A Clark converter unit (8) and No. two Park inverse transformation block (9), for by the positional information of rotor described three-phase current signal i _a, i _bcarry out successively Clark conversion and Park inverse transformation, obtain d shaft current value of feedback i _dwith q shaft current value of feedback i _q;

Band pass filter BPF (10), for to described three-phase current signal i _a, i _bfiltering, obtains particular harmonic current i _am, i _bm;

No. two Clark converter units (11), for to described particular harmonic current i _am, i _bmcarry out Clark conversion, obtain the current i under alpha-beta axle system _{α m}, i _{β m};

Position detection device unit (12), for the current i under to described alpha-beta axle being _{α m}, i _{β m}carry out signal processing, obtain rotor feedback angular velocity omega _rand the positional information of rotor

2. a high-power internal permanent magnet synchronous motor method for controlling position-less sensor, is characterized in that, the method comprises the following steps:

Step 1, angular speed with rotor feedback angular velocity omega _rit is poor to do, and this difference is carried out to PI computing and obtain given torque

Step 2, to described given torque carry out the computing of MTPA breakdown torque current ratio, obtain d, q shaft current set-point

Step 3, to q shaft current set-point with q shaft current value of feedback i _qdifference carry out PI computing, obtain q shaft voltage set-point to d shaft current set-point with d shaft current value of feedback i _ddifference carry out PI computing, obtain d shaft voltage set-point

Step 4, to described q shaft voltage set-point d shaft voltage set-point positional information with rotor carry out Park inverse transformation, obtain respectively the given voltage of α axle with the given voltage of β axle

Step 5, according to the given voltage of described α axle with the given voltage of β axle obtain modulation ratio, carry out three level SHMPWM computings, and then the pulse width signal of generating power device;

Step 6, export the m subharmonic voltage vector u that contains constant amplitude, 120 ° of phase angles of phase phasic difference by three-level inverter unit according to described power device pulse width signal _am, u _bm, u _cmthree-phase tri-level voltage signal, this three-phase tri-level voltage signal is added on three phase windings of high-power PMSM, realizes the vector control of PMSM, on motor winding, produces three-phase current signal, and exports the current i of a phase and b phase _a, i _b;

Step 7, current i to a phase described in step 6 and b phase _a, i _bcarry out successively Clark conversion and Park inverse transformation, obtain d shaft current value of feedback i _dwith q shaft current value of feedback i _q;

Step 8, the current i of employing band pass filter BPF to described a phase and b phase _a, i _bfiltering, obtains particular harmonic current i _am, i _bm;

Step 9, to described particular harmonic current i _am, i _bmcarry out Clark conversion, obtain the current i under alpha-beta axle system _{α m}, i _{β m};

Step 10, position detection device unit are to the current i under described alpha-beta axle system _{α m}, i _{β m}carry out signal processing, obtain rotor feedback angular velocity omega _rand the positional information of rotor complete once high-power internal permanent magnet synchronous motor position Sensorless Control, and return to execution step one, carry out high-power internal permanent magnet synchronous motor position Sensorless Control next time.

3. the high-power internal permanent magnet synchronous motor method for controlling position-less sensor of one according to claim 2, is characterized in that, described in step 5 according to the given voltage of described α axle with the given voltage of β axle obtain modulation ratio, carry out three level SHMPWM computings, and then export the m subharmonic voltage vector u that contains constant amplitude, 120 ° of phase angles of phase phasic difference by three-level inverter unit according to described power device pulse width signal described in the process of the pulse width signal of generating power device and step 6 _am, u _bm, u _cmthe acquisition process of three-phase tri-level voltage signal as follows:

Step one by one, be located at and in [0, pi/2] the individual cycle, have N half-convergency, a _i(i=1,2,3 ..., N) and be half-convergency, decompose the amplitude H of inverter output voltage base component of degree n n according to Fourier series ₁and the amplitude H of j subharmonic composition _jwith half-convergency a _ibetween relational expression be:

$H_1=\frac{4}{\mathrm{\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^i\sin \left({\mathrm{ja}}_i\right)\right]---\left(2\right)$

$H_j=\frac{4}{\mathrm{j\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^i\sin \left({\mathrm{ja}}_i\right)\right]---\left(3\right);$

Step 1 two, set up the non-linear transcendental equation group of SHMPWM modulation strategy,

|M _a-H ₁|≤L ₁

$\frac{1}{\left|H_1\right|}\frac{4}{\mathrm{j\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^i\sin \left({\mathrm{ja}}_i\right)\right]\≤L_j$

$\frac{1}{\left|H_1\right|}\frac{4}{\mathrm{m\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^i\sin \left({\mathrm{ja}}_i\right)\right]{=L}_m---\left(4\right);$

Wherein M _afor modulation ratio; J=5,7,9 ..., m-2, m+2 ...; M is the number of times of realizing the required particular harmonic of position Sensorless Control and become sub-signal, and m is odd number; L _j, L _mthe harmonic content value allowing under IEEE519-1992 standard for N-1 kind particular harmonic composition;

Step 1 three, by the solving of the non-linear transcendental equation group in step 1 two, solution procedure adopts the discrete method solving of optimization, obtains different modulating and compares M _athe position of the switching angle of the N in situation, and make the discrete question blank of SHMPWM algorithm;

Step 1 four, utilize the given voltage of α axle with the given voltage of β axle calculate the size of required modulation ratio, the switching angle position of SHMPWM algorithm while searching this state in the discrete question blank in step 1 three, described switching angle position is the pulse width signal of power device; Produce three level voltage signals according to this three-level inverter unit, switching angle position, in this voltage signal, comprise the m subharmonic voltage vector u of constant amplitude _am, u _bm, u _cm, N-1 kind harmonic wave has disappeared simultaneously.

4. the high-power internal permanent magnet synchronous motor method for controlling position-less sensor of one according to claim 2, is characterized in that, in step 10, position observer unit is to the current i under described alpha-beta axle system _{α m}, i _{β m}carry out signal processing, obtain rotor feedback angular velocity omega _rand the positional information of rotor concrete steps comprise:

Steps A, the particular harmonic composition current model of structure internal permanent magnet synchronous motor under alpha-beta axle system;

Step B, in internal permanent magnet synchronous motor model, introduce the static reference frame γ-δ of two-phase axle system, obtain the particular harmonic composition current model under γ-δ axle system;

Wherein, γ-δ axle system closes with the coordinate of alpha-beta axle system and is: γ axle is ahead of 45 °, α axle, and δ axle is ahead of 90 °, γ axle;

Step C, structure three-phase m subharmonic voltage equation, and select m subharmonic voltage as particular harmonic voltage;

Step D, utilize principle of coordinate transformation and the cosine law, obtain the m subharmonic composition current envelope curve equation under alpha-beta and γ-δ axle are according to the particular harmonic composition current model under the lower particular harmonic composition current model of the alpha-beta axle system in steps A, γ-δ axle system in step B and the three-phase m subharmonic voltage equation in step C;

Step e, go out rotor-position according to described m subharmonic composition current envelope curve equation inference under alpha-beta and γ-δ axle system estimation equation;

Step F, to described rotor-position carry out differential, obtain rotor feedback angular velocity omega _r;

Step G, general _eand ω _rfeed back to control system, realize the control to position-sensor-free motor speed.

5. the high-power internal permanent magnet synchronous motor method for controlling position-less sensor of one according to claim 2, is characterized in that, the particular harmonic composition current model under alpha-beta axle system in steps A is:

$\left[\begin{array}{c}{i}_{\mathrm{\αm}}\\ i_{\mathrm{\βm}}\end{array}\right]=\left[\begin{array}{cc}{L}_0-L_1\cos \left(2{\mathrm{\θ}}_e\right)& -L_1\sin \left({2\mathrm{\θ}}_e\right)\\ -L_1\sin \left({2\mathrm{\θ}}_e\right)& L_0+L_1\cos \left({2\mathrm{\θ}}_e\right)\end{array}\right]\·\frac{1}{L_0^2-L_1^2}\·\left[\begin{array}{c}{\∫u}_{\mathrm{\αm}}\mathrm{dt}\\ \∫u_{\mathrm{\βm}}\mathrm{dt}\end{array}\right]$

Wherein, c represents radio-frequency component component; L ₀for average inductance, L ₀=(L _d+ L _q)/2; L ₁be half poor inductance, L ₁=(L _d-L _q)/2; L _d, L _qbe respectively d axle inductive component and the q axle inductive component of motor; θ _efor the space potential angle setting between stator a phase axis and rotor d axle.

6. the high-power internal permanent magnet synchronous motor method for controlling position-less sensor of one according to claim 2, is characterized in that, the particular harmonic composition current model under the γ-δ axle system in step B is:

$\left[\begin{array}{c}{i}_{\mathrm{\γm}}\\ i_{\mathrm{\δm}}\end{array}\right]=\left[\begin{array}{cc}{L}_0-L_1\sin \left(2{\mathrm{\θ}}_e\right)& L_1\cos \left({2\mathrm{\θ}}_e\right)\\ L_1\cos \left({2\mathrm{\θ}}_e\right)& L_0+L_1\sin \left({2\mathrm{\θ}}_e\right)\end{array}\right]\·\frac{1}{L_0^2-L_1^2}\·\left[\begin{array}{c}{\∫u}_{\mathrm{\γm}}\mathrm{dt}\\ \∫u_{\mathrm{\δm}}\mathrm{dt}\end{array}\right]---\left(8\right).$

7. the high-power internal permanent magnet synchronous motor method for controlling position-less sensor of one according to claim 2, is characterized in that, builds m subharmonic voltage equation in step C, and selects m subharmonic voltage as follows as the selection foundation of particular harmonic voltage:

According to Fourier analysis, the voltage expression formula of the three-phase m subharmonic voltage composition of three-level inverter output is:

u _am＝H _msin(mωt)

u _bm＝H _msin(mωt-m×120°)

u _cm＝H _msin(mωt+m×120°) (5)；

Wherein, H _mfor m subharmonic becomes the amplitude of component voltage, H _mwith M _a, L _mand three-level inverter DC bus-bar voltage U _dcrelevant, H _m=M _al _mu _dc/ 2;

Select m=3k+1, k=1,2,3 ..., formula (5) can be expressed as:

u _am＝H _msin(mωt)

u _bm＝H _msin(mωt-k×360°-120°)＝H _msin(mωt-120°) (6)

u _cm＝H _msin(mωt+k×360°+120°)＝H _msin(mωt+120°)

By formula (5), select m=3k+1, k=1,2,3 ... when subharmonic, the amplitude that can know this m subharmonic is constant, the phase place of the m subharmonic voltage of a phase, b phase and c phase differs the electrical degree of 120 ° in space, as becoming sub-signal for the particular harmonic of rotor position information estimation, and learn thus m ∈ { 7,13,19......}.

8. the high-power internal permanent magnet synchronous motor method for controlling position-less sensor of one according to claim 2, is characterized in that, the m subharmonic composition current envelope curve equation under alpha-beta and γ-δ axle system in step D is:

M subharmonic composition current envelope curve equation under alpha-beta axle system is:

$\left[\begin{array}{c}{\left|i_{\mathrm{\αm}}\right|}_{\mathrm{peak}}\\ {\left|i_{\mathrm{\βm}}\right|}_{\mathrm{peak}}\end{array}\right]=\frac{3\×H_m}{2\×\mathrm{m\ω}(L_0^2-L_1^2)}\left[\begin{array}{c}\sqrt{L_0^2+L_1^2+{2L}_0{L}_1\cos \left({2\mathrm{\θ}}_e\right)}\\ \sqrt{L_0^2+L_1^2-{2L}_0{L}_1\cos \left({2\mathrm{\θ}}_e\right)}\end{array}\right];$

M subharmonic composition current envelope curve equation under γ-δ axle system is:

$\left[\begin{array}{c}{\left|i_{\mathrm{\γc}}\right|}_{\mathrm{peak}}\\ {\left|i_{\mathrm{\δc}}\right|}_{\mathrm{peak}}\end{array}\right]=\frac{3\×H_m}{2\×\mathrm{m\ω}(L_0^2-L_1^2)}\left[\begin{array}{c}\sqrt{L_0^2+L_1^2+{2L}_0{L}_1\sin \left({2\mathrm{\θ}}_e\right)}\\ \sqrt{L_0^2+L_1^2-{2L}_0{L}_1\sin \left({2\mathrm{\θ}}_e\right)}\end{array}\right].$

9. the high-power internal permanent magnet synchronous motor method for controlling position-less sensor of one according to claim 2, is characterized in that,

In step e, go out rotor-position according to described m subharmonic composition current envelope curve equation inference under alpha-beta and γ-δ axle system the concrete steps of estimation equation comprise:

M subharmonic composition current envelope curve equation under steps A 1, α axle is:

${{\left|i_{\mathrm{\αm}}\right|}_{\mathrm{peak}}}^2=\frac{9\×{H_m}^2}{4\×m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}[L_0^2+L_1^2+{2L}_0{L}_1\cos \left(2{\mathrm{\θ}}_e\right)];$

M subharmonic composition current envelope curve equation under steps A 2, β axle is:

${{\left|i_{\mathrm{\βm}}\right|}_{\mathrm{peak}}}^2=\frac{9\×{H_m}^2}{4\×m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}[L_0^2+L_1^2-{2L}_0{L}_1\cos \left(2{\mathrm{\θ}}_e\right)];$

Steps A 3, by steps A 1 | i _{α m}| _peak ²with steps A 2 | i _{β m}| _peak ²bring following formula into and calculate,

${{\left|i_{\mathrm{\αm}}\right|}_{\mathrm{peak}}}^2-{{\left|i_{\mathrm{\βm}}\right|}_{\mathrm{peak}}}^2=\frac{{{9H}_m}^2{L}_0{L}_1}{m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}\cos \left({2\mathrm{\θ}}_e\right);$

M subharmonic composition current envelope curve equation under steps A 4, γ axle is:

${{\left|i_{\mathrm{\γm}}\right|}_{\mathrm{peak}}}^2=\frac{9\×{H_m}^2}{4\×m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}[L_0^2+L_1^2+{2L}_0{L}_1\sin \left(2{\mathrm{\θ}}_e\right)];$

M subharmonic composition current envelope curve equation under steps A 5, δ axle is:

${{\left|i_{\mathrm{\δm}}\right|}_{\mathrm{peak}}}^2=\frac{9\×{H_m}^2}{4\×m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}[L_0^2+L_1^2-{2L}_0{L}_1\sin \left(2{\mathrm{\θ}}_e\right)];$

Steps A 6, by steps A 4 | i _{γ m}| _peak ²with steps A 5 | i _{δ m}| _peak ²bringing following formula into calculates:

${{\left|i_{\mathrm{\γm}}\right|}_{\mathrm{peak}}}^2-{{\left|i_{\mathrm{\δm}}\right|}_{\mathrm{peak}}}^2=\frac{{{9H}_m}^2{L}_0{L}_1}{m^2{\mathrm{\ω}}^2{(L_0^2-L_1^2)}^2}\sin \left({2\mathrm{\θ}}_e\right);$

Steps A 7, bring steps A 3 into following formula with the value in steps A 6 and calculate:

$\frac{{{\left|i_{\mathrm{\γm}}\right|}_{\mathrm{peak}}}^2-{{\left|i_{\mathrm{\δm}}\right|}_{\mathrm{peak}}}^2}{{{\left|i_{\mathrm{\αm}}\right|}_{\mathrm{peak}}}^2-{{\left|i_{\mathrm{\βm}}\right|}_{\mathrm{peak}}}^2}=\frac{\sin \left({2\mathrm{\θ}}_e\right)}{\cos \left({2\mathrm{\θ}}_e\right)}=\tan \left({2\mathrm{\θ}}_e\right);$

Steps A 8, can obtain rotor-position by steps A 7 estimation equation be