CN102638216B - Method for starting motor without position sensor - Google Patents

Abstract

The invention discloses a method for starting a motor without a position sensor, which includes: applying a fixed voltage vector to the motor and obtaining Is; subjecting the Is to Clark conversion and obtaining i <alpha> and i <beta>; presetting rotation speed omega r and torque T<e> as 0 by computing to obtain rotor flux-linkage vector angle theta r, subjecting omega r and omega r to PI operation to obtain Te, then subjecting Te and the T3 to PI operation to obtain delta, obtaining theta<s> according to the formula theta <s>=theta <r> +delta, giving Psi* <s> to obtain U<alpha> and U<beta>, subjecting the U<alpha> and the U<beta> to inverse Clark conversion and obtaining U<a>, U<b> and U<c>, acquiring I<a>, I<b> and I<c> through current, subjecting the I<a>, the I<b> and the I<c> to Clark conversion and obtaining I alpha and I beta, obtaining Psi <s> alpha and Psi<s> beta by stator flux linkage operation, obtaining Psi<s> and theta<s> by modular angle operation, obtaining the Te by torque estimate, discretizing the theta<s> to obtain omega<r>, and finishing motor starting process if the omega<r> is equal to the omega*<r>. The method for starting the motor without the position sensor has the following advantages that by combining initial position detection technology with DTC (direct torque control) strategy, loop-locked stable control of the integral control strategy is realized.

Description

Position-sensor-free electric motor starting method

Technical field

The invention belongs to electronic engineering technical field, particularly a kind of initial position detection method of utilizing is position-sensor-free electric motor starting method in conjunction with the advanced electric motor starting method that does not need position transducer of Strategy of Direct Torque Control.

Background technology

Space vector pulse width modulation Space Vector Pulse Width Modulation abbreviation: SVPWM

Direct torque control Direct Torque Control abbreviation: DTC

Field orientation control Field Oriented Control abbreviation: FOC

Integration scale operation Proportion Integral abbreviation: PI

Although the directed FOC strategy of controlling of existing motor-field is without position transducer, but in the time of electric motor starting, need dragging motor to arrive after a certain rotating speed, just can make motor speed adjustable continuously, that is to say and in motor start-up procedure, adopt Speed open-loop control strategy, when motor accelerates to after the rotating speed of setting, be switched to speed closed loop control strategy, extended the start-up time of motor; In order to overcome the defect that in field orientation control FOC strategy, electric motor starting needs open loop to drag, there is motor direct torque control DTC strategy, in the time of electric motor starting, do not need dragging motor to arrive a certain rotating speed, but need to be used in conjunction with position transducer or velocity transducer, therefore in use there will be following problem:

1) in design process, need to increase the hardware circuit supporting with transducer, cause the raising of design cost;

2) transducer hardware support kit circuit is easy to be subject to electromagnetic interference, causes signal transmission errors, and then control strategy was lost efficacy;

3) be subject to the restriction in the useful life of transducer itself, need to regularly replace transducer, in the time of emat sensor more, need dismantle whole motor encapsulation, more change jobs loaded down with trivial details.

Summary of the invention

Technical problem to be solved by this invention is, provides a kind of initial position detection technique and DTC control strategy to combine, and makes whole control strategy reach the position-sensor-free electric motor starting method of the stable control of closed loop.

Space vector pulse width modulation SVPWM algorithm, for according to voltage vector, adopts Using dSPACE of SVPWM algorithm generating power device pulse width signal;

Voltage source inverter, for producing three-phase winding current according to described power device pulse width signal, sends to permagnetic synchronous motor;

Clark conversion, fastens the coordinate transform of three-phase winding current on two phase coordinate system α, β by static coordinate.

For solving the problems of the technologies described above, position-sensor-free electric motor starting method provided by the invention, comprises the following steps:

(1) apply fixed voltage vector V to motor and a is switched on mutually, b, c phase no power and another fixed voltage vector V ' to a phase no power, b, c switch on mutually, then obtain respectively I by voltage source inverter _s, I ' _sif, I _s-I ' _s> 0 so-π/6 < θ _r< π/6 if I _s-I ' _s< 0, so 5 π/6 < θ _r< 7 π/6 ${\mathrm{\&theta;}}_r=\frac{1}{2}\arctan [\frac{i_{\mathrm{\&beta;}}}{(i_{\mathrm{\&alpha;}}-I_s)}+2\mathrm{\&pi;}];$

(2) to I _scarry out Clark conversion, obtain the current i in two phase coordinate systems _α, i _β, by calculating $\left[\begin{array}{c}{i}_{\mathrm{\&alpha;}}\\ i_{\mathrm{\&beta;}}\end{array}\right]=\sqrt{\frac{2}{3}}\frac{V_{\mathrm{DC}}}{R}*\left[\begin{array}{c}1-\frac{1}{2}(e^{-\frac{R}{\mathrm{Lq}}\mathrm{Ts}}+e^{-\frac{R}{\mathrm{Ld}}\mathrm{Ts}})+\frac{1}{2}(e^{-\frac{R}{\mathrm{Lq}}\mathrm{Ts}}-e^{-\frac{R}{\mathrm{Ld}}\mathrm{Ts}})\cos 2{\mathrm{\&theta;}}_r\\ \frac{1}{2}(e^{-\frac{R}{\mathrm{Lq}}\mathrm{Ts}}+e^{-\frac{R}{\mathrm{Ld}}\mathrm{Ts}})\sin 2{\mathrm{\&theta;}}_r\end{array}\right]=\left[\begin{array}{c}{I}_s+{\mathrm{\&Delta;I}}_s\cos 2{\mathrm{\&theta;}}_r\\ {\mathrm{\&Delta;I}}_s\sin 2{\mathrm{\&theta;}}_r\end{array}\right]$ ?

To rotor flux azimuth θ _r;

Wherein: i _α, i _βfor known quantity; V _dCfor DC bus-bar voltage; R is armature winding resistance; Ld is d axle inductance; Lq is q axle inductance; Ts applies the time that the voltage vector time apply V and V '; E is mathematical constant; Δ I _sfor bus current variable quantity (is I _s, I ' _sdifference).

(3) given rotating speed of target right and ω _rcarry out PI computing and obtain target torque again to T and carry out after PI computing, obtaining angle of torsion δ, now ω _r=0, T _e=0;

Wherein: PI computing is used for described difference and ω _rcarry out ratio, integral operation obtains target torque to described difference T _ewith carry out ratio, integral operation obtains angle of torsion δ.

(4) by rotor flux azimuth θ _rwith stator magnetic linkage azimuth θ _sconversion formula θ _s=θ _r+ δ, obtains stator magnetic linkage azimuth θ _s;

(5) give again the magnetic linkage that sets the goal carry out voltage vector computing $\begin{array}{c}{u}_{\mathrm{\&alpha;}}=i_{\mathrm{\&alpha;}}{r}_s+\frac{\left|{\mathrm{\&psi;}}_s^{*}\right|\cos ({\mathrm{\&theta;}}_s+\mathrm{\&delta;})-\left|{\mathrm{\&psi;}}_s\right|\cos {\mathrm{\&theta;}}_s}{\mathrm{\&Delta;T}}\\ u_{\mathrm{\&beta;}}=i_{\mathrm{\&beta;}}{r}_s+\frac{\left|{\mathrm{\&psi;}}_s^{*}\right|\sin ({\mathrm{\&theta;}}_s+\mathrm{\&delta;})-\left|{\mathrm{\&psi;}}_s\right|\sin {\mathrm{\&theta;}}_s}{\mathrm{\&Delta;T}}\end{array}$

Obtain the voltage U in two phase coordinate systems _α, U _β;

Wherein: i _α, i _β, δ, θ _sfor known; r _sfor stator resistance; Δ T is sampling time (equating with Ts).(6) to U _α, U _βcarry out contrary Clark conversion, obtain voltage U a, Ub, Uc in three phase coordinate systems, adopt by electric current

Collection obtains three-phase current I _a, I _b, I _c;

Two phase coordinate system α, the upper winding voltage of β are transformed to three phase coordinate system a, the upper winding voltage of b, c by contrary Clark conversion.

(7) to I _a, I _b, I _ccarry out Clark conversion, obtain current I _α, I _β, calculate by stator magnetic linkage $\begin{array}{c}{\mathrm{\&psi;}}_{\mathrm{s\&alpha;}}=\&Integral;(u_{\mathrm{\&alpha;}}-i_{\mathrm{\&alpha;}}{r}_s)\mathrm{dt}\\ {\mathrm{\&psi;}}_{\mathrm{s\&beta;}}=\&Integral;(u_{\mathrm{\&beta;}}-i_{\mathrm{\&beta;}}{r}_s)\mathrm{dt}\end{array}$ Obtain the component Ψ of current stator magnetic linkage _{s α}, ψ _{s β};

Wherein: r _sfor stator resistance.

$\left|{\mathrm{\&psi;}}_s\right|=\sqrt{{\mathrm{\&psi;}}_{\mathrm{s\&alpha;}}^2+{\mathrm{\&psi;}}_{\mathrm{s\&beta;}}^2}$

(8) by modular angle computing θ _s=arctan (ψ _{s β}/ Ψ _{s α}) draw the mould Ψ of current stator magnetic linkage _sand current stator magnetic linkage azimuth θ _s;

(9) pass through torque estimating draw current torque T _e, and to θ _scarry out discretization and obtain current rotational speed omega _r;

Wherein: p is magnetic pole logarithm.

(10) if motor start-up procedure finishes; If repeating step (3)～step (9), now step (3) medium speed ω _rwith torque T _efor the rotational speed omega calculating in step (9) _rwith torque T _e, until the rotational speed omega calculating in step (9) _rwith rotating speed of target equate.

Further, the described current acquisition of step (6) refers to according to voltage vector and adopts space vector pulse width modulation SVPWM algorithm generating power device pulse width signal, by voltage source inverter, described power device pulse width signal is produced to three-phase winding current again, send to permagnetic synchronous motor PMSM.

Adopt after above structure, position-sensor-free electric motor starting method of the present invention compared with prior art, has the following advantages:

1, without position transducer, calculate the initial position of rotor with initial position detection algorithm, need not consider signal transmission errors completely and the loss that causes, and then avoid the control strategy loss causing out of control, improve stability and reliability;

2, electric motor starting and accelerator are all under closed-loop control state, have shortened motor starting time, improved operating efficiency;

3, greatly reduce design cost, reduce energy resource consumption.

Brief description of the drawings

Fig. 1 is the block diagram of position-sensor-free electric motor starting method of the present invention.

Embodiment

Below by embodiment, the invention will be further described by reference to the accompanying drawings.

As shown in Figure 1, first, the initial position that utilizes initial position detection method to calculate rotor is θ _r, concrete grammar is: given voltage vector V, obtains three-phase total current I by voltage source inverter _s, then to I _scarry out Clark conversion, obtain i _α, i _β,

Pass through formula $\left[\begin{array}{c}{i}_{\mathrm{\&alpha;}}\\ i_{\mathrm{\&beta;}}\end{array}\right]=\sqrt{\frac{2}{3}}\frac{V_{\mathrm{DC}}}{R}*\left[\begin{array}{c}1-\frac{1}{2}(e^{-\frac{R}{\mathrm{Lq}}\mathrm{Ts}}+e^{-\frac{R}{\mathrm{Ld}}\mathrm{Ts}})+\frac{1}{2}(e^{-\frac{R}{\mathrm{Lq}}\mathrm{Ts}}-e^{-\frac{R}{\mathrm{Ld}}\mathrm{Ts}})\cos 2{\mathrm{\&theta;}}_r\\ \frac{1}{2}(e^{-\frac{R}{\mathrm{Lq}}\mathrm{Ts}}+e^{-\frac{R}{\mathrm{Ld}}\mathrm{Ts}})\sin 2{\mathrm{\&theta;}}_r\end{array}\right]$

After arrangement, obtain $\left[\begin{array}{c}{i}_{\mathrm{\&alpha;}}\\ i_{\mathrm{\&beta;}}\end{array}\right]=\left[\begin{array}{c}{I}_s+{\mathrm{\&Delta;I}}_s\cos 2{\mathrm{\&theta;}}_r\\ {\mathrm{\&Delta;I}}_s\sin 2{\mathrm{\&theta;}}_r\end{array}\right]$

Calculate ${\mathrm{\&theta;}}_r=\frac{1}{2}\arctan \frac{i_{\mathrm{\&beta;}}}{\left(i_{\mathrm{\&alpha;}}{-I}_s\right)}$ Or ${\mathrm{\&theta;}}_r=\frac{1}{2}\arctan [\frac{i_{\mathrm{\&beta;}}}{(i_{\mathrm{\&alpha;}}-I_s)}+2\mathrm{\&pi;}],$

A given voltage vector V ', obtains I ' by voltage source inverter again _s, relatively I _s, I ' _ssize,

If I _s-I ' _s> 0, so-π/6 < θ _r< π/6 ${\mathrm{\&theta;}}_r=\frac{1}{2}\arctan \frac{i_{\mathrm{\&beta;}}}{\left(i_{\mathrm{\&alpha;}}{-I}_s\right)};$

If I _s-I ' _s< 0, so 5 π/6 < θ _r< 7 π/6 ${\mathrm{\&theta;}}_r=\frac{1}{2}\arctan [\frac{i_{\mathrm{\&beta;}}}{(i_{\mathrm{\&alpha;}}-I_s)}+2\mathrm{\&pi;}],$

Then, obtain current rotational speed omega by Strategy of Direct Torque Control _r, concrete grammar is as follows:

1) when the rotational speed omega of front motor _rwith torque T _ebe 0, given rotating speed of target right and ω _rcarry out obtaining target torque after PI computing again to T _ewith carry out obtaining angle of torsion δ after PI computing;

2) by rotor flux azimuth θ _rwith stator magnetic linkage azimuth θ _sconversion formula θ _s=θ _r+ δ, obtains stator magnetic linkage azimuth θ _s;

3) giving the magnetic linkage that sets the goal carry out voltage vector formula $\begin{array}{c}{u}_{\mathrm{\&alpha;}}=i_{\mathrm{\&alpha;}}{r}_s+\frac{\left|{\mathrm{\&psi;}}_s^{*}\right|\cos ({\mathrm{\&theta;}}_s+\mathrm{\&delta;})-\left|{\mathrm{\&psi;}}_s\right|\cos {\mathrm{\&theta;}}_s}{\mathrm{\&Delta;T}}\\ u_{\mathrm{\&beta;}}=i_{\mathrm{\&beta;}}{r}_s+\frac{\left|{\mathrm{\&psi;}}_s^{*}\right|\sin ({\mathrm{\&theta;}}_s+\mathrm{\&delta;})-\left|{\mathrm{\&psi;}}_s\right|\sin {\mathrm{\&theta;}}_s}{\mathrm{\&Delta;T}}\end{array}$ Calculate the voltage U in two phase coordinate systems _α, U _β;

4) to U _α, U _βcarry out contrary Clark conversion, obtain voltage U a, Ub, Uc in three phase coordinate systems, obtain three-phase current I by SVPWM algorithm and voltage source inverter _a, I _b, I _c;

5) to I _a, I _b, I _ccarry out Clark conversion, obtain the I on current two phase coordinate system α, β _α, I _β, by stator magnetic linkage formula $\begin{array}{c}{\mathrm{\&psi;}}_{\mathrm{s\&alpha;}}=\&Integral;(u_{\mathrm{\&alpha;}}-i_{\mathrm{\&alpha;}}{r}_s)\mathrm{dt}\\ {\mathrm{\&psi;}}_{\mathrm{s\&beta;}}=\&Integral;(u_{\mathrm{\&beta;}}-i_{\mathrm{\&beta;}}{r}_s)\mathrm{dt}\end{array}$

Calculate the component Ψ of current stator magnetic linkage _{s α}, ψ _{s β};

$\left|{\mathrm{\&psi;}}_s\right|=\sqrt{{\mathrm{\&psi;}}_{\mathrm{s\&alpha;}}^2+{\mathrm{\&psi;}}_{\mathrm{s\&beta;}}^2}$

6) by modular angle operational formula θ _s=arctan (ψ _{s β}/ Ψ _{s α}) draw the mould Ψ of current stator magnetic linkage _sand current stator magnetic linkage azimuth θ _s;

7) again by torque estimating formula $T_e=\frac{3}{2}p({\mathrm{\&psi;}}_{\mathrm{s\&alpha;}}{i}_{\mathrm{\&beta;}}-{\mathrm{\&psi;}}_{\mathrm{s\&beta;}}{i}_{\mathrm{\&alpha;}})$ Draw current torque T _e, and to θ _scarry out discretization and obtain current rotational speed omega _r;

8) if motor start-up procedure finishes;

If repeating step (1)～step (7), now motor is worked, therefore step (1) medium speed ω _rwith torque T _efor the rotational speed omega calculating in step (7) _rwith torque T _e, until the rotational speed omega calculating in step (7) _rwith rotating speed of target equate.

So far this position-sensor-free motor start-up procedure finishes, and the motor operation after making reaches stable, closed-loop control fast.

Claims (8)

1. a position-sensor-free electric motor starting method, is characterized in that, comprises the following steps:

(1) apply fixed voltage vector to motor, then obtain three-phase total current I by voltage source inverter _s;

(2) to three-phase total current I _scarry out Clark conversion, obtain the current i in two phase coordinate systems _α, i _β, by calculating rotor flux azimuth θ _r, concrete formula is:

$\left[\begin{array}{c}{i}_{\mathrm{\&alpha;}}\\ i_{\mathrm{\&beta;}}\end{array}\right]=\sqrt{\frac{2}{3}}\frac{V_{\mathrm{DC}}}{R}*\left[\begin{array}{c}1-\frac{1}{2}(e^{-\frac{R}{\mathrm{Lq}}\mathrm{Ts}}+e^{-\frac{R}{\mathrm{Ld}}\mathrm{Ts}})+\frac{1}{2}(e^{-\frac{R}{\mathrm{Lq}}\mathrm{Ts}}-e^{-\frac{R}{\mathrm{Ld}}\mathrm{Ts}})\cos 2{\mathrm{\&theta;}}_r\\ \frac{1}{2}(e^{-\frac{R}{\mathrm{Lq}}\mathrm{Ts}}+e^{-\frac{R}{\mathrm{Ld}}\mathrm{Ts}})\sin 2{\mathrm{\&theta;}}_r\end{array}\right]=\left[\begin{array}{c}{I}_s+{\mathrm{\&Delta;I}}_s\cos 2{\mathrm{\&theta;}}_r\\ {\mathrm{\&Delta;I}}_s\sin {2\mathrm{\&theta;}}_r\end{array}\right]$

${\mathrm{\&theta;}}_r=\frac{1}{2}\arctan \frac{i_{\mathrm{\&beta;}}}{(i_{\mathrm{\&alpha;}}-\mathrm{Is})}$ Or ${\mathrm{\&theta;}}_r=\frac{1}{2}\arctan [\frac{i_{\mathrm{\&beta;}}}{(i_{\mathrm{\&alpha;}}-I_s)}+2\mathrm{\&pi;}]$

Wherein: V _dCfor DC bus-bar voltage; R is armature winding resistance; Ld is d axle inductance; Lq is q axle inductance; Ts applies the time that the voltage vector time apply V and V '; E is mathematical constant; △ I _sfor bus current variable quantity is I _s, I' _sdifference;

Initial position detection method: first pass into voltage vector V and a is switched on mutually, b, c phase no power, obtain I by current acquisition _s, then passing into another voltage vector V ' to a phase no power, b, c switch on mutually, obtain I' _sif, Is-I' _s>0 so-π/6< θ _r< π/6 if I _s-I' _s<0,5 π so/6< θ _r<7 π/6

(3) given rotating speed of target now motor does not start, rotational speed omega _rwith torque T _ebe 0, right and ω _rcarry out obtaining target torque after integration scale operation again to T _ewith carry out obtaining angle of torsion δ after integration scale operation; Described integration scale operation is used for difference and ω _rcarry out ratio, integral operation obtains target torque to difference T _ewith carry out ratio, integral operation obtains angle of torsion δ;

(4) by rotor flux azimuth θ _rwith stator magnetic linkage azimuth θ _sconversion formula θ _s=θ _r+ δ, obtains stator magnetic linkage azimuth θ _s;

(5) give again the magnetic linkage that sets the goal by the i that step (1)～(4) calculate _α, i _β, δ, θ _scarry out voltage vector computing and obtain the voltage U in two phase coordinate systems _α, U _β;

(6) to the voltage U in two phase coordinate systems _α, U _βcarry out contrary Clark conversion, obtain voltage U a, Ub, Uc in three phase coordinate systems, obtain three-phase current I by current acquisition _a, I _b, I _c;

(7) to I _a, I _b, I _ccarry out Clark conversion, obtain the electric current I in two current phase coordinate systems _α, I _β, calculate the component ψ of current stator magnetic linkage by stator magnetic linkage _{s α}, ψ _{s β};

(8) draw the mould ψ of current stator magnetic linkage by modular angle computing _sand current stator magnetic linkage azimuth θ _s;

(9) draw current torque T by torque estimating _e, and to θ _scarry out discretization and obtain current rotational speed omega _r;

(10) if motor start-up procedure finishes; If repeating step (3)～step (9), now step (3) medium speed ω _rwith torque T _efor the rotational speed omega calculating in step (9) _rwith torque T _e, until the rotational speed omega calculating in step (9) _rwith rotating speed of target equate.

2. position-sensor-free electric motor starting method according to claim 1, is characterized in that, the Clark conversion that step (2) and step (7) are described, fastens the coordinate transform of three-phase winding current on two phase coordinate system α, β by static coordinate; Two phase coordinate system α, the upper winding voltage of β are transformed to three phase coordinate system a, the upper winding voltage of b, c by the described contrary Clark conversion of step (6).

3. position-sensor-free electric motor starting method according to claim 1, it is characterized in that, the described current acquisition of step (6) refers to according to voltage vector and adopts space vector pulse width modulation SVPWM algorithm generating power device pulse width signal, by voltage source inverter, described power device pulse width signal is produced to three-phase winding current again, send to permagnetic synchronous motor PMSM.

4. position-sensor-free electric motor starting method according to claim 3, is characterized in that, described space vector pulse width modulation SVPWM algorithm, for according to voltage vector, adopts Using dSPACE of SVPWM algorithm generating power device pulse width signal; Voltage source inverter, for producing three-phase winding current according to described power device pulse width signal, sends to permagnetic synchronous motor.

5. position-sensor-free electric motor starting method according to claim 1, is characterized in that, the described voltage vector operational formula of step (5) is $\begin{array}{c}{u}_{\mathrm{\&alpha;}}=i_{\mathrm{\&alpha;}}{r}_s+\frac{\left|{\mathrm{\&psi;}}_s^{*}\right|\cos ({\mathrm{\&theta;}}_s+\mathrm{\&delta;})-\left|{\mathrm{\&psi;}}_s\right|\cos {\mathrm{\&theta;}}_s}{\mathrm{\&Delta;T}}\\ u_{\mathrm{\&beta;}}=i_{\mathrm{\&beta;}}{r}_s+\frac{\left|{\mathrm{\&psi;}}_s^{*}\right|\sin ({\mathrm{\&theta;}}_s+\mathrm{\&delta;})-\left|{\mathrm{\&psi;}}_s\right|\sin {\mathrm{\&theta;}}_s}{\mathrm{\&Delta;T}}\end{array},$

Wherein: r _sfor stator resistance; △ T equates with Ts in the sampling time.

6. position-sensor-free electric motor starting method according to claim 1, is characterized in that, the described stator magnetic linkage computing formula of step (7) is $\begin{array}{c}{\mathrm{\&psi;}}_{\mathrm{s\&alpha;}}=\&Integral;(u_{\mathrm{\&alpha;}}-i_{\mathrm{\&alpha;}}{r}_s)\mathrm{dt}\\ {\mathrm{\&psi;}}_{\mathrm{s\&beta;}}=\&Integral;(u_{\mathrm{\&beta;}}-i_{\mathrm{\&beta;}}{r}_s)\mathrm{dt}\end{array},$

Wherein: r _sfor stator resistance.

7. position-sensor-free electric motor starting method according to claim 1, is characterized in that, the described modular angle operational formula of step (8) is $\begin{array}{c}\left|{\mathrm{\&psi;}}_s\right|=\sqrt{{\mathrm{\&psi;}}_{\mathrm{s\&alpha;}}^2+{\mathrm{\&psi;}}_{\mathrm{s\&beta;}}^2}\\ {\mathrm{\&theta;}}_s=\arctan ({\mathrm{\&psi;}}_{\mathrm{s\&beta;}}/{\mathrm{\&psi;}}_{\mathrm{s\&alpha;}})\end{array}.$

8. position-sensor-free electric motor starting method according to claim 1, is characterized in that, the described torque estimating formula of step (9) is $T_e=\frac{3}{2}p({\mathrm{\&psi;}}_{\mathrm{s\&alpha;}}{i}_{\mathrm{\&beta;}}-{\mathrm{\&psi;}}_{\mathrm{s\&beta;}}{i}_{\mathrm{\&alpha;}}),$

Wherein: p is magnetic pole logarithm.