CN104410255A - Construction method of Fourier equation for three-level selected harmonic elimination pulse width modulation - Google Patents

Abstract

The invention belongs to the field of variable frequency modulation of alternating current inverters. In order to ensure that selected harmonic elimination pulse width modulation achieves an expected harmonic elimination effect, the technical scheme adopted by the invention is that a construction method of a Fourier equation for three-level selected harmonic elimination pulse width modulation comprises the following steps: 1, deriving a midpoint voltage fluctuation equation and carrying out quantitative analysis on voltages at the two ends of a support capacitor; 2, through analyzing a midpoint voltage expression, knowing and controlling a moderate amount of triple frequency harmonic waves contained in output phase voltage to restrain midpoint voltage fluctuation; 3, reconstructing the Fourier equation as shown in the specification; Udc is direct current side voltage; N is the number of switch angles within 1/4 period; g=2,4,6,8,10; ak is an unknown switch angle. The construction method is mainly used for the variable frequency modulation of the alternating current inverters.

Description

Three level particular harmonic eliminates the building method of modulation Fourier equation group

Technical field

The invention belongs to AC inverter Frequency-variable Modulation field, relate to the correlation technique of field of power electronics.Specifically, the building method that three level particular harmonic eliminates modulation Fourier equation group is related to.

Technical background

Particular harmonic eliminate modulation (SHEPWM) due to its switching frequency low, the advantages such as output harmonic wave content is little, be widely used in high-power frequency conversion modulation field.Development along with power electronic technology and the continuous lifting to converter capacity requirement, three level particular harmonic is eliminated modulation technique and is widely used with NPC type tri-level inversion structure, and topological structure as shown in Figure 1.Compared to common inversion modulation strategy, three level particular harmonic eliminates modulation can ensure outputting high quality waveform under lower switching frequency; The features such as inverter voltage endurance capability is strong.

Under NPC type inverter circuit, carry out three level particular harmonic eliminate modulation, serious mid-point voltage fluctuation problem can be caused.The method of current this problem of solution is main mainly with control strategy, adds the complexity of control system.

Summary of the invention

For overcoming the deficiencies in the prior art, make particular harmonic eliminate modulation and reach expection harmonic elimination effect, the technical scheme that the present invention takes is, three level particular harmonic eliminates the building method of modulation Fourier equation group, comprises the following steps:

One, derive mid-point voltage wave equation, quantitative analysis carried out to Support Capacitor both end voltage:

(1) arranging output phase voltage angle value is θ; I _mfor fundamental current amplitude; M is modulation ratio; C is Support Capacitor value; ω is angular speed, is divided into six regions a cycle, and will produce mid point electric current as three-phase output current access mid point O, then expression formula is:

$i_o=\left\{\begin{array}{c}-i_a{S}_a+i_b{S}_b-i_c{S}_c,\mathrm{\θ}\∈[0,\frac{\mathrm{\π}}{3})\\ -i_a{S}_a+i_b{S}_b+i_c{S}_c,\mathrm{\θ}\∈[\frac{\mathrm{\π}}{3},\frac{2\mathrm{\π}}{3})\\ -i_a{S}_a-i_b{S}_b+i_c{S}_c,\mathrm{\θ}\∈[\frac{2\mathrm{\π}}{3},\mathrm{\π})\\ i_a{S}_a-i_b{S}_b+i_c{S}_c,\mathrm{\θ}\∈[\mathrm{\π},\frac{4\mathrm{\π}}{3})\\ i_a{S}_a-i_b{S}_b-i_c{S}_c,\mathrm{\θ}\∈[\frac{4\mathrm{\π}}{3},\frac{5\mathrm{\π}}{3})\\ i_a{S}_a+i_b{S}_b-i_c{S}_c,\mathrm{\θ}\∈[\frac{5\mathrm{\π}}{3},2\mathrm{\π})\end{array}\right.$

In formula, i _ofor mid point electric current; i _a, i _b, i _cfor three-phase current; S _a, S _b, S _cfor on off state function;

Convert on off state function to continuous function, only consider interim first-harmonic and triple-frequency harmonics, then mid point current expression is

$i_o=\left\{\begin{array}{c}{i}_{o1}+K\·i_b\left(\mathrm{\τ}\right),\mathrm{\θ}\∈[0+\frac{2\mathrm{\π}}{3}l,\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\\ i_{o2}-K\·i_a\left(\mathrm{\τ}\right),\mathrm{\θ}\∈[\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l,\frac{2\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\end{array}\right.$

In formula, l=0,1,2; K=2Mk _h3sin (3 θ); τ=θ-2 π l/3; for the angle of the delayed phase voltage of phase current; i _a(τ) represent a phase current i _athe function that translation 2 π l/3 phase place obtains, other phase implication is roughly the same;

(2) alignment electric current carries out integral operation, and the capacitance voltage that is supported expresses formula, u _c1for single Support Capacitor voltage, i.e. mid-point voltage; u ^* _c1for first-harmonic effect produces voltage; u _h3for triple-frequency harmonics effect produces voltage; Mid-point voltage formula is as follows:

$u_{c1}=u_{c1}^{*}+u_{h3}$

It is as follows that first-harmonic effect produces voltage expression:

Because formula is longer, for ease of writing, with the subitem in A, B acute pyogenic infection of finger tip above formula, wherein,

It is as follows that triple-frequency harmonics effect produces voltage expression

$u_{h3}=\frac{I_m{\mathrm{Mk}}_{h3}}{4\mathrm{C\ω}}\left\{\begin{array}{c}[D-J-F],\mathrm{\θ}\∈[0+\frac{2\mathrm{\π}}{3}l,\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\\ [E+F],\mathrm{\θ}\∈[\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l,\frac{2\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\end{array}\right.$

Because formula is longer, for ease of writing, with the subitem in D, E, F, J acute pyogenic infection of finger tip above formula, wherein, k _h3for triple-frequency harmonics and the ratio of fundamental voltage amplitude;

Two, can know that controlling to contain appropriate triple-frequency harmonics in output phase voltage can suppress mid-point voltage to fluctuate by analyzing mid-point voltage expression formula: calculated respectively by output phase voltage pulse amplitude, according to each magnitude of voltage size, weights coefficient is set, be incorporated in the middle of Fourier equation group, with u _akthe approximate amplitude as the pulse of a phase voltage kth, definition pulse ripple weights FACTOR P _i, i=1,2 ..., N, and work as time, there is P _2k-1=P _2k=u _k+ u _n-k+1when time, have $P_N=u_{{\left(\frac{N+1}{2}\right)}_L}+u_{{\left(\frac{N+1}{2}\right)}_R};$

Three, Fourier equation group is reconstructed as follows

$\left\{\begin{array}{c}\frac{4}{\mathrm{\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^{k+1}\frac{P_i}{U_{\mathrm{dc}}}\cos {\mathrm{\α}}_k\right]=M\\ \frac{4}{3\mathrm{\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^{k+1}\frac{P_i}{U_{\mathrm{dc}}}\cos {3\mathrm{\α}}_k\right]={\mathrm{Mk}}_{h3}\\ \underset{k=1}{\overset{N}{\mathrm{\Σ}}}{(-1)}^{k+1}{P}_i\cos q{\mathrm{\α}}_k=0,q=\mathrm{5,7},...,3g\±1\end{array}\right.$

In formula, U _dcfor DC voltage; N is switching angle number in 1/4th cycles; G=2,4,6,8,10; α _kfor switching angle to be asked.

Compared with the prior art, technical characterstic of the present invention and effect:

The three level particular harmonic provided according to the present invention is eliminated modulation Fourier equation group and is carried out switching angle and solve, and gained switching angle can suppress NPC inverter mid-point voltage fluctuate, reduction output line voltage low-order harmonic.And the present invention realizes simply, not increasing system hardware structure.

Accompanying drawing explanation

Fig. 1 is that particular harmonic eliminates modulation realization flow figure.

Fig. 2 is threephase switch view.

Fig. 3 is 1/4th cycle N number of switching angle phase voltage timing charts.Due to mid-point voltage fluctuation problem, in figure, not etc., magnitude of voltage is not respectively u to pulse voltage amplitude height ₁, u ₂u _n.

Fig. 4 flow chart of the present invention.

Embodiment

The present invention is directed to the angle of this problem from modulation algorithm, propose a kind of brand-new Fourier equation set constructor method.The Fourier equation group utilizing the present invention to construct is carried out switching angle and is solved, and calculated switching angle can suppress mid-point voltage to fluctuate, and improves harmonic suppression effect.

The present invention gives three level particular harmonic and eliminate the method that in modulation, Fourier equation group re-constructs, technical scheme is as follows:

1. derive mid-point voltage wave equation, carry out quantitative analysis to Support Capacitor both end voltage, step is as follows:

(1) arranging output phase voltage angle value is θ; I _mfor fundamental current amplitude; M is modulation ratio; C is Support Capacitor value; ω is angular speed.Is divided into six regions one cycle, as shown in Figure 2, will produce mid point electric current as three-phase output current access mid point O, then expression formula is

$i_o=\left\{\begin{array}{c}-i_a{S}_a+i_b{S}_b-i_c{S}_c,\mathrm{\θ}\∈[0,\frac{\mathrm{\π}}{3})\\ -i_a{S}_a+i_b{S}_b+i_c{S}_c,\mathrm{\θ}\∈[\frac{\mathrm{\π}}{3},\frac{2\mathrm{\π}}{3})\\ -i_a{S}_a-i_b{S}_b+i_c{S}_c,\mathrm{\θ}\∈[\frac{2\mathrm{\π}}{3},\mathrm{\π})\\ i_a{S}_a-i_b{S}_b+i_c{S}_c,\mathrm{\θ}\∈[\mathrm{\π},\frac{4\mathrm{\π}}{3})\\ i_a{S}_a-i_b{S}_b-i_c{S}_c,\mathrm{\θ}\∈[\frac{4\mathrm{\π}}{3},\frac{5\mathrm{\π}}{3})\\ i_a{S}_a+i_b{S}_b-i_c{S}_c,\mathrm{\θ}\∈[\frac{5\mathrm{\π}}{3},2\mathrm{\π})\end{array}\right.$

In formula, i _ofor mid point electric current; i _a, i _b, i _cfor three-phase current; S _a, S _b, S _cfor on off state function.

Convert on off state function to continuous function, only consider interim first-harmonic and triple-frequency harmonics, then mid point current expression is

$i_o=\left\{\begin{array}{c}{i}_{o1}+K\·i_b\left(\mathrm{\τ}\right),\mathrm{\θ}\∈[0+\frac{2\mathrm{\π}}{3}l,\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\\ i_{o2}-K\·i_a\left(\mathrm{\τ}\right),\mathrm{\θ}\∈[\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l,\frac{2\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\end{array}\right.$

In formula, l=0,1,2; K=2Mk _h3sin (3 θ); τ=θ-2 π l/3; for the angle of the delayed phase voltage of phase current; i _a(τ) represent a phase current i _athe function that translation 2 π l/3 phase place obtains, other phase implication is identical.

(2) alignment electric current carries out integral operation, and the capacitance voltage that just can be supported expresses formula.U _c1for single Support Capacitor voltage, i.e. mid-point voltage; u ^* _c1for first-harmonic effect produces voltage; u _h3for triple-frequency harmonics effect produces voltage; Mid-point voltage formula is as follows:

$u_{c1}=u_{c1}^{*}+u_{h3}$

It is as follows that first-harmonic effect produces voltage expression:

Because formula is longer, for ease of writing, with the subitem in A, B acute pyogenic infection of finger tip above formula, wherein,

It is as follows that triple-frequency harmonics effect produces voltage expression:

$u_{h3}=\frac{I_m{\mathrm{Mk}}_{h3}}{4\mathrm{C\ω}}\left\{\begin{array}{c}[D-J-F],\mathrm{\θ}\∈[0+\frac{2\mathrm{\π}}{3}l,\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\\ [E+F],\mathrm{\θ}\∈[\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l,\frac{2\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\end{array}\right.$

Because formula is longer, for ease of writing, with the subitem in D, E, F, J acute pyogenic infection of finger tip above formula, wherein, k _h3for triple-frequency harmonics and the ratio of fundamental voltage amplitude.

2. can know that controlling to contain appropriate triple-frequency harmonics in output phase voltage can suppress mid-point voltage to fluctuate by analyzing mid-point voltage expression formula.Output phase voltage pulse amplitude is calculated respectively, as shown in Figure 3.Weights coefficient is set according to each magnitude of voltage size, is incorporated in the middle of Fourier equation group.With u _akthe approximate amplitude as a kth pulse, definition pulse ripple weights FACTOR P i, i=1,2 ..., N, and work as time, there is P _2k-1=P _2k=u _k+ u _n-k+1when time, have $P_N=u_{{\left(\frac{N+1}{2}\right)}_L}+u_{{\left(\frac{N+1}{2}\right)}_R}.$

3. the present invention adds the appropriate control to triple-frequency harmonics in original Fourier equation group, can effectively suppress mid-point voltage to fluctuate.Pulse amplitude fluctuation is considered in the middle of switching angle solving equation simultaneously, be equivalent to revise switching angle according to pulse actual voltage value, to reduce the impact that mid-point voltage fluctuation produces modulation strategy, improve the inhibition for low-order harmonic.Reconstruct Fourier equation group is as follows

$\left\{\begin{array}{c}\frac{4}{\mathrm{\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^{k+1}\frac{P_i}{U_{\mathrm{dc}}}\cos {\mathrm{\α}}_k\right]=M\\ \frac{4}{3\mathrm{\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^{k+1}\frac{P_i}{U_{\mathrm{dc}}}\cos {3\mathrm{\α}}_k\right]={\mathrm{Mk}}_{h3}\\ \underset{k=1}{\overset{N}{\mathrm{\Σ}}}{(-1)}^{k+1}{P}_i\cos q{\mathrm{\α}}_k=0,q=\mathrm{5,7},...,3g\±1\end{array}\right.$

In formula, U _dcfor DC voltage; N is switching angle number in 1/4th cycles; G=2,4,6,8,10; α _kfor switching angle to be asked.

The three level particular harmonic provided according to the present invention is eliminated modulation Fourier equation group and is carried out switching angle and solve, and gained switching angle can suppress NPC inverter mid-point voltage fluctuate, reduction output line voltage low-order harmonic.And the present invention realizes simply, not increasing system hardware structure.

The present invention is further described below in conjunction with specific embodiment.

1. determine phase voltage pulse position and amplitude

Three level particular harmonic in the present invention is eliminated modulation and conventional modulated method first will be utilized to obtain switching angle.According to the phase voltage waveform figure that switching angle position and load are determined as shown in Figure 3.Calculate each pulse voltage amplitude of a phase u _{a k} , computing formula is as follows

$u_{c1}=u_{c1}^{*}+u_{h3}$

In formula, first-harmonic effect generation voltage fluctuation computing formula is as follows

In formula, the partial expression in A, B acute pyogenic infection of finger tip formula, l=0,1,2;

It is as follows that triple-frequency harmonics effect produces voltage fluctuation computing formula

$u_{h3}=\frac{I_m{\mathrm{Mk}}_{h3}}{4\mathrm{C\ω}}\left\{\begin{array}{c}[D-J-F],\mathrm{\θ}\∈[0+\frac{2\mathrm{\π}}{3}l,\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\\ [E+F],\mathrm{\θ}\∈[\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l,\frac{2\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\end{array}\right.$

In formula, the partial expression in D, E, F, J acute pyogenic infection of finger tip formula, k _h3for triple-frequency harmonics and the ratio of fundamental voltage amplitude.

2. pulse ripple weights coefficient calculations

Obtain each pulse center point magnitude of voltage u _klaggard horizontal pulse fluctuation weights FACTOR P _icalculate.When time, P _2k-1=P _2k=u _k+ u _n-k+1.When time, triple-frequency harmonics is controlled 16% for fundamental voltage amplitude, by k simultaneously ₁₃be set to 0.16, fluctuate in order to suppress mid-point voltage.

3. calculated the coefficient of variation P in Fourier equation group before cosine term by step 2 _i, just can construct novel Fourier equation group as shown in the formula

$\left\{\begin{array}{c}\frac{4}{\mathrm{\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^{k+1}\frac{P_i}{U_{\mathrm{dc}}}\cos {\mathrm{\α}}_k\right]=M\\ \frac{4}{3\mathrm{\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^{k+1}\frac{P_i}{U_{\mathrm{dc}}}\cos {3\mathrm{\α}}_k\right]={\mathrm{Mk}}_{h3}\\ \underset{k=1}{\overset{N}{\mathrm{\Σ}}}{(-1)}^{k+1}{P}_i\cos q{\mathrm{\α}}_k=0,q=\mathrm{5,7},...,3g\±1\end{array}\right.$

Novel Fourier equation group considers voltage fluctuation to the impact exporting pulse, also adds the control to phase voltage triple-frequency harmonics simultaneously, makes it also have a small amount of triple-frequency harmonics.What can be optimized modulation strategy switching angle by the Fourier equation group newly built solves work.

4. using the initial disaggregation of conventional modulated switching angle as Fourier equation group, utilize that newton's pressgang is inferior carries out iterative computation, then namely the switching angle newly calculated can be used as the foundation exporting PWM waveform.The switching angle utilizing the Fourier equation group of neotectonics to solve can effectively suppress NPC type three-level inverter mid-point voltage to fluctuate, and reduces fluctuation and particular harmonic is eliminated to the impact of modulating and causing, improve harmonic inhibition capability.

Claims (1)

1. three level particular harmonic eliminates a building method for modulation Fourier equation group, it is characterized in that, comprises the following steps:

One, derive mid-point voltage wave equation, quantitative analysis carried out to Support Capacitor both end voltage:

(1) arranging output phase voltage angle value is θ; I _mfor fundamental current amplitude; M is modulation ratio; C is Support Capacitor value; ω is angular speed, is divided into six regions a cycle, and will produce mid point electric current as three-phase output current access mid point O, then expression formula is:

$i_o=\left\{\begin{array}{c}-i_a{S}_a+i_b{S}_b-i_c{S}_c,\mathrm{\θ}\∈[0,\frac{\mathrm{\π}}{3})\\ -i_a{S}_a+i_b{S}_b+i_c{S}_c,\mathrm{\θ}\∈[\frac{\mathrm{\π}}{3},\frac{2\mathrm{\π}}{3})\\ -i_a{S}_a-i_b{S}_b+i_c{S}_c,\mathrm{\θ}\∈[\frac{2\mathrm{\π}}{3},\mathrm{\π})\\ i_a{S}_a-i_b{S}_b+i_c{S}_c,\mathrm{\θ}\∈[\mathrm{\π},\frac{4\mathrm{\π}}{3})\\ i_a{S}_a-i_b{S}_b-i_c{S}_c,\mathrm{\θ}\∈[\frac{4\mathrm{\π}}{3},\frac{5\mathrm{\π}}{3})\\ i_a{S}_a+i_b{S}_b-i_c{S}_c,\mathrm{\θ}\∈[\frac{5\mathrm{\π}}{3},2\mathrm{\π})\end{array}\right.$

In formula, i _ofor mid point electric current; i _a, i _b, i _cfor three-phase current; S _a, S _b, S _cfor on off state function;

Convert on off state function to continuous function, only consider interim first-harmonic and triple-frequency harmonics, then mid point current expression is

$i_o=\left\{\begin{array}{cc}{i}_{o1}+K\·i_b\left(\mathrm{\τ}\right),& \mathrm{\θ}\∈[0+\frac{2\mathrm{\π}}{3}l,\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\\ i_{o2}-K\·i_a\left(\mathrm{\τ}\right),& \mathrm{\θ}\∈[\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l,\frac{2\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\end{array}\right.$

In formula, l=0,1,2; K=2Mk _h3sin (3 θ); τ=θ-2 π l/3; for the angle of the delayed phase voltage of phase current; i _a(τ) represent a phase current i _athe function that translation 2 π l/3 phase place obtains, other phase implication is roughly the same;

(2) alignment electric current carries out integral operation, and the capacitance voltage that is supported expresses formula, u _c1for single Support Capacitor voltage, i.e. mid-point voltage; u ^* _c1for first-harmonic effect produces voltage; u _h3for triple-frequency harmonics effect produces voltage; Mid-point voltage formula is as follows:

$u_{c1}=u_{c1}^{*}+u_{h3}$

It is as follows that first-harmonic effect produces voltage expression:

Because formula is longer, for ease of writing, with the subitem in A, B acute pyogenic infection of finger tip above formula, wherein, l=0,1,2;

It is as follows that triple-frequency harmonics effect produces voltage expression

$u_{h3}=\frac{I_m{\mathrm{Mk}}_{h3}}{4\mathrm{C\ω}}\left\{\begin{array}{c}[D-J-F],\mathrm{\θ}\∈[0+\frac{2\mathrm{\π}}{3}l,\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\\ [E+F],\mathrm{\θ}\∈[\frac{\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l,\frac{2\mathrm{\π}}{3}+\frac{2\mathrm{\π}}{3}l)\end{array}\right.$

Because formula is longer, for ease of writing, with the subitem in D, E, F, J acute pyogenic infection of finger tip above formula, wherein, k _h3for triple-frequency harmonics and the ratio of fundamental voltage amplitude;

Two, can know that controlling to contain appropriate triple-frequency harmonics in output phase voltage can suppress mid-point voltage to fluctuate by analyzing mid-point voltage expression formula: calculated respectively by output phase voltage pulse amplitude, according to each magnitude of voltage size, weights coefficient is set, be incorporated in the middle of Fourier equation group, with u _akthe approximate amplitude as the pulse of a phase voltage kth, definition pulse ripple weights FACTOR P _i, i=1,2 ..., N, and work as k=1, time, there is P _2k-1=P _2k=u _k+ u _n-k+1when time, have $P_N=u_{{\left(\frac{N+1}{2}\right)}_L}+u_{{\left(\frac{N+1}{2}\right)}_R};$

Three, Fourier equation group is reconstructed as follows

$\left\{\begin{array}{c}\frac{4}{\mathrm{\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^{k+1}\frac{P_i}{U_{\mathrm{dc}}}\cos {\mathrm{\α}}_k\right]=M\\ \frac{4}{3\mathrm{\π}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left[{(-1)}^{k+1}\frac{P_i}{U_{\mathrm{dc}}}\cos 3{\mathrm{\α}}_k\right]={\mathrm{Mk}}_{h3}\\ \underset{k=1}{\overset{N}{\mathrm{\Σ}}}{(-1)}^{k+1}{P}_i\cos q{\mathrm{\α}}_k=0,q=\mathrm{5,7},...,3g\±1\end{array}\right.$

In formula, U _dcfor DC voltage; N is switching angle number in 1/4th cycles; G=2,4,6,8,10; α _kfor switching angle to be asked.