CN104362841B - A kind of 18 pulse wave rectifier system harmonicses suppression systems and method - Google Patents

Abstract

A kind of 18 pulse wave rectifier system harmonicses suppression systems and method, belong to commutation system harmonics restraint system and method.Pass through single-phase electricity die mould inverter, exchange auxiliary square-wave voltage is produced to act on main circuit DC inductance, improve diode rectifier bridge output current wave, by rectifier transformer primary side and secondary side current relation, so as to reach the purpose for suppressing current on line side harmonic wave.Advantage：1. the auxiliary circuit is single-phase electricity die mould inverter, topological structure is simple, with low cost；2. by the size for controlling ac square-wave voltage on DC inductance so that three three-phase diode rectifier bridges can load-sharing electric current, while the linear raising and lowering of diode rectifier bridge output current, improves DC side current waveform；3. auxiliary circuit is by transformer coupled on main circuit, auxiliary circuit power is smaller, and power device switching loss is smaller；4. current on line side harmonic wave is 1.42%, current waveform is sufficiently close to 1 close to sine wave, power factor.

Description

A kind of 18 pulse wave rectifier system harmonicses suppression systems and method

Technical field

The present invention relates to a kind of commutation system harmonics restraint system and method, particularly a kind of 18 pulse wave rectifier system harmonicses Suppression system and method.

Background technology

With the development of industrial technology, power electronic devices is widely used in power rectifier industry, but its rectification Contain substantial amounts of non-linear electrical load in system, bring substantial amounts of harmonic pollution to power network, cause a series of problem： (1) some of harmonic wave composition may cause power system local parallel resonant so that harmonic content is amplified, and causes capacitor Damage；(2) substantial amounts of harmonic wave and reactive power occupy many capacity of system, while also increasing electric power in transmitting procedure In loss, cause the increase of system cost；(3) to its exterior, harmonic wave will launch the electromagnetic wave of powerful corresponding frequencies, Serious interference is caused to electronic equipment and communication equipment；(4) harmonic wave reduces the generation, transmission and utilization ratio of electric energy, makes Device of overheating of electrical, generation vibration and noise, and make insulation ag(e)ing, service life shortens, or even breaks down or burn；(5) Harmonic wave deteriorates power network electric circumstance, it is also possible to causes automatics and protective relaying device misoperation, fills electric energy metrical Put and mistake occur, trigger unnecessary trouble.

The method of suppression system harmonic wave mainly has two kinds in multi-pulse rectification system, and one kind is humorous by being installed to system Ripple compensation device carrys out the harmonic wave of compensation system generation, and another is to use multi-pulse rectification technology to multi phase rectifier.Pass through The system of giving installing harmonic compensation device, the harmonic wave that this kind of method is produced to various harmonic sources plays the role of good, for example, install Active or passive filter come compensate or harmonic carcellation source produce harmonic wave.But multi phase rectifier is commonly used to The occasion of high-power, and the power grade and the power grade of multi-pulse rectification system of wave filter be more or less the same, then choosing The cost of system can not only be increased by selecting the method for this harmonics restraint, increase the loss of system, while adding power device Quantity, causing the stability of system reduces, and the method for this harmonics restraint is passive-type in addition, be harmonic source produce it is humorous Just start to suppress harmonic wave after ripple, this when, harmonic wave exerted a certain influence to system.Another method, to many Pulses Rectifier uses multi-pulse rectification technology, and this kind of method be divided into two classes, and one kind is increase system pulse number, so as to reduce and be System harmonic wave, but increase pulse number can increase rectifier transformer manufacture difficulty, increase volume, price is uprised；It is another to be pair Transformed on the basis of circuit system, carry out current-modulation in DC side, so as to improve input current waveform, be close to just String ripple, suppresses current on line side harmonic wave.This kind of method relative to pulsewidth modulation (Pulse Width Modulation, PWM) technology, Cheap with simple in construction, high reliability is widely used to aircraft rectifier and D.C. high voltage transmission etc. big In power rectification system.

18 traditional impulse commutation modes can eliminate 5,7,11,13 subharmonic, but its input current in input current Total harmonic distortion (Total Harmonic distortion, THD) be 9.78%, harmonic pollution is still larger, it is impossible to meet The requirement of green energy resource.

The content of the invention

The invention aims to provide a kind of 18 pulse wave rectifier system harmonicses suppression system and method, traditional 18 are solved Pulse wave rectifier system causes the problem of system bulk and capacity increase in AC using active filter compensation harmonic electric current.

The object of the present invention is achieved like this：18 pulse wave rectifiers carry out harmonics restraint in DC side includes system and side Method；Described harmonics restraint system includes：18 pulse wave rectifier transformers, 3 three-phase diode uncontrollable rectifier bridges, 3 high frequencies become Depressor, 3 DC inductances, 3 groups of voltage-type single-phase inversion circuits, voltage sensor A, voltage sensor B, voltage sensor C, electricity Pressure sensor D and current sensor E, control circuit and drive circuit；

18 pulse wave rectifier transformer inputs get access to grid, 3 groups of output ends of 18 pulse wave rectifier transformers respectively with 3 three-phases Diode uncontrollable rectifier bridge input is connected, and the output end of 3 three-phase diode uncontrollable rectifier bridges simultaneously couples load；Voltage-type list Phase inverter circuit DC side is in parallel with system load, and output end is coupled on main circuit by high frequency transformer；Voltage sensor A, voltage sensor B and voltage sensor C sample the output voltages of 3 three-phase diode rectifier bridges respectively, voltage sensor D and Current sensor E sampling systems load side voltage and electric current, voltage sensor A, voltage sensor B, voltage sensor C, voltage The result of sensor D and current sensor E samplings is input to control circuit input end, control circuit output end and drive circuit Input is connected, and drive circuit exports switching device of the pwm pulse to single-phase inversion circuit.

Diode rectifier bridge output voltage sensor and load side voltage sensor outlet side connection control circuit, draw output Voltage alternating component；Load-side current sensor outlet side is connected with control circuit, is multiplied with the ac square wave signal of definition, Obtain exchange auxiliary square-wave voltage；Both are added and obtain single-phase inverter modulated signal by control circuit, with triangle carrier signal Compare and obtain PWM drive signal, and PWM drive signal is sent to drive circuit；Drive circuit, which drives the PWM received, to be believed Export after number carrying out power amplification and give voltage-type single-phase inversion circuit；Voltage-type single-phase inversion circuit direct side and system load side Parallel connection, outlet side will exchange auxiliary square-wave voltage by high frequency transformer and be coupled on DC inductance, so as to change DC side electricity It is linear rise or linear decline to flow waveform, reaches the purpose for suppressing rectifier transformer current on line side harmonic wave.

Described harmonic suppressing method：Using auxiliary circuit voltage-type single-phase inverter, output produces ac square-wave voltage, By transformer coupled on main circuit DC inductance so that rectifier bridge output current linear change, improve rectifier bridge output electricity Waveform is flowed, suppresses rectifier transformer current on line side harmonic wave；The harmonic suppression technique comprises the following steps：

(1) AC compounent Δ U acquisition：By taking first group of rectifier bridge as an example, diode rectifier bridge output voltage U_rec1For 6 ripples The DC voltage of head, therefore its AC compounent Δ U=U_rec1-U_o；

(2) ac square-wave voltage u_iAcquisition：In order to ensure diode output current linear rise and decline, and three-phase two Pole pipe rectification circuit energy load-sharing electric current, by taking first group of rectifier bridge as an example, as 30 ° of 0 ＜ ω t ＜, square-wave voltage u_iWidth Value U_S=-L₁di_rec1/ dt, commutation in order to eliminate diode rectifier bridge is delayed and load-sharing electric current, it is necessary in ω t=30 ° When force i_rec1For 0, and i_rec1Average value beThen：

Wherein f is 50Hz, L₁For REC1 DC side inductance, I_LFor load current；

(3) auxiliary circuit modulating waveAcquisition：By main circuit AC compounent Δ U^*With ac square-wave voltageIt is added and makees For modulating wave, i.e.,

(4) auxiliary circuit output voltage U₁Acquisition：Auxiliary circuit switch device drive signal is auxiliary circuit modulating waveObtained compared with high frequency carrier, auxiliary circuit DC side is in parallel with main circuit load-side, obtain output voltage U₁=Δ U+ u_i；

(5) rectifier transformer net side A phase currents I_aAcquisition：3 two poles are drawn according to the principle of 18 pulse wave rectifier systems 20 ° are differed in pipe rectifier bridge output current wave phase, the expression formula of three groups of electric currents of DC side is write out, calculates rectifier transformer Net side A phase currents I_aExpression formula, verify invention achieves suppress side harmonics purpose.

Beneficial effect, as a result of such scheme, has realized in 18 pulse wave rectifier system dc lateral inhibition AC Harmonics of Input, by the control to single-phase inversion circuit output voltage, obtains an exchange on main circuit DC inductance Aid in square-wave voltage so that diode rectifier bridge output current linearly rises or falls, and improves DC side current waveform, profit With the relation between the current harmonics of alternating current-direct current side, to influence the input current of AC, so as to inhibit AC input current Harmonic wave, it is achieved thereby that 18 pulse wave rectifier systems DC side to AC carry out harmonics restraint purpose.

Compared with the method for existing harmonics restraint, the present invention has advantages below：1. the auxiliary circuit is single-phase voltage Type inverter, topological structure is simple, it is easy to accomplish 18 traditional pulse wave rectifier circuits are improved, with low cost；2. pass through Control the size of ac square-wave voltage on DC side inductance so that three three-phase fully-controlled diode rectifier bridges being capable of load-sharing electricity Stream；3. auxiliary circuit is by transformer coupled on main circuit, the transformer primary secondary turn ratio is 20:1, auxiliary circuit power ratio compared with Small, power device switching loss is smaller；4. current on line side harmonic wave is 1.42%, current waveform close to sine wave, and power because Number is sufficiently close to 1, and output voltage is identical with 18 traditional pulse wave rectifier circuit output voltage waveforms.

Brief description of the drawings

Fig. 1 is 18 pulse wave rectifier harmonics restraint system main circuit topology figures.

Fig. 2 is extended triangle phase-shifting mode figure.

Fig. 3 is single-phase inverter topological diagram.

Fig. 4 is first group of rectifier bridge Rec1 fundamental diagram.

Fig. 5 is 3 groups of diode rectifier bridge output current wave figures.

Fig. 6 is rectifier transformer primary side current oscillogram.

Fig. 7 is rectifier transformer primary side current simulation waveform.

Fig. 8 is rectifier transformer primary side A phase current simulation waveforms.

Fig. 9 is rectifier transformer primary side A phase current fft analysis figures.

Embodiment

18 pulse wave rectifiers of the invention carry out harmonics restraint in DC side includes system and method；Described harmonics restraint system Including：18 pulse wave rectifier transformers, respectively 3 three-phase diode uncontrollable rectifier bridges, diode uncontrollable rectifier bridge Rec1, two poles Pipe uncontrollable rectifier bridge Rec2, diode uncontrollable rectifier bridge Rec3,3 high frequency transformers, respectively high frequency transformer T1, high frequency become Depressor T2 and high frequency transformer T3, respectively 3 DC inductances, DC inductance L₁, DC inductance L₂With DC inductance L₃, 3 groups of electricity Die mould single-phase inversion circuit, voltage sensor A, voltage sensor B, voltage sensor C, voltage sensor D and current sensor E, control circuit and drive circuit.

18 pulse wave rectifier transformer inputs get access to grid, 3 groups of output ends of 18 pulse wave rectifier transformers respectively with 3 three-phases Diode uncontrollable rectifier bridge input is connected, and the output end of 3 three-phase diode uncontrollable rectifier bridges simultaneously couples load；Voltage-type list Phase inverter circuit DC side is in parallel with system load, and output end is coupled on main circuit by high frequency transformer；Voltage sensor A, voltage sensor B and voltage sensor C sample the output voltages of 3 three-phase diode rectifier bridges respectively, voltage sensor D and Current sensor E sampling systems load side voltage and electric current, voltage sensor A, voltage sensor B, voltage sensor C, voltage The result of sensor D and current sensor E samplings is input to control circuit input end, control circuit output end and drive circuit Input is connected, and drive circuit exports switching device of the pwm pulse to single-phase inversion circuit.

The no-load voltage ratio of described high frequency transformer is 20:1, working frequency is 300Hz, high pressure side joint voltage-type single-phase inversion electricity Road output end, low pressure side joint main circuit, the voltage that single-phase inverter is exported is by transformer access major loop, making direct current Voltage on inductance is ac square-wave voltage.

When the voltage on DC inductance is ac square-wave voltage, then inductive current linearly changes, linear rise or decline, Improve DC side current waveform.

Described voltage-type single-phase inversion circuit direct side is in parallel with system load side, and outlet side passes through high frequency transformer coupling Close on main circuit.

Described rectifier bridge output voltage sensor is used to detect rectifier bridge output voltage, rectifier bridge output voltage sensor The input of output end connection control circuit, for compared with DC side load voltage sensor, obtaining exporting AC compounent Size.

DC side load current sensor is used to detect DC side load current, DC side load current sensor output end With controlling the input of circuit to be connected, the size for calculating ac square-wave voltage amplitude.

The input of control circuit is connected with the output end of 5 sensors, the output end and drive circuit of control circuit Input is connected, and drive circuit output pwm pulse controls the work of single-phase inversion circuit.

Described harmonic suppressing method：Using auxiliary circuit voltage-type single-phase inverter, output produces ac square-wave voltage, By transformer coupled on main circuit DC inductance so that rectifier bridge output current linear change, improve rectifier bridge output electricity Waveform is flowed, suppresses rectifier transformer current on line side harmonic wave；The harmonic suppression technique comprises the following steps：

(1) AC compounent Δ U acquisition：By taking first group of rectifier bridge as an example, diode rectifier bridge output voltage U_rec1For 6 ripples The DC voltage of head, therefore its AC compounent Δ U=U_rec1-U_o；

(2) ac square-wave voltage u_iAcquisition：In order to ensure diode output current linear rise and decline, and three-phase two Pole pipe rectification circuit energy load-sharing electric current, by taking first group of rectifier bridge as an example, as 30 ° of 0 ＜ ω t ＜, square-wave voltage u_iWidth Value U_S=-L₁di_rec1/ dt, commutation in order to eliminate diode rectifier bridge is delayed and load-sharing electric current, it is necessary in ω t=30 ° When force i_rec1For 0, and i_rec1Average value beThen：

Wherein f is 50Hz, L₁For REC1 DC side inductance, I_LFor load current；

(3) auxiliary circuit modulating waveAcquisition：By main circuit AC compounent Δ U^*With ac square-wave voltageIt is added and makees For modulating wave, i.e.,

(4) auxiliary circuit output voltage U₁Acquisition：Auxiliary circuit switch device drive signal is auxiliary circuit modulating waveObtained compared with high frequency carrier, auxiliary circuit DC side is in parallel with main circuit load-side, obtain output voltage U₁=Δ U+ u_i；

(5) rectifier transformer net side A phase currents I_aAcquisition：3 two poles are drawn according to the principle of 18 pulse wave rectifier systems 20 ° are differed in pipe rectifier bridge output current wave phase, the expression formula of three groups of electric currents of DC side is write out, calculates rectifier transformer Net side A phase currents I_aExpression formula, it is suppressed that side harmonics.

Diode rectifier bridge output voltage sensor and load side voltage sensor outlet side connection control circuit, draw output Voltage alternating component；Load-side current sensor outlet side is connected with control circuit, is multiplied with the ac square wave signal of definition, Obtain exchange auxiliary square-wave voltage；Both are added and obtain single-phase inverter modulated signal by control circuit, with triangle carrier signal Compare and obtain PWM drive signal, and PWM drive signal is sent to drive circuit；

The PWM drive signal received export after power amplification by drive circuit gives voltage-type single-phase inversion circuit；

Voltage-type single-phase inversion circuit direct side is in parallel with system load side, and outlet side is by high frequency transformer by AC auxiliary Help square-wave voltage to be coupled on DC inductance, so as to change DC side current waveform for linear rise or linear decline, reach suppression The purpose of rectifier transformer current on line side harmonic wave processed.

Embodiment：A kind of method that 18 pulse wave rectifier system harmonicses suppress, it is characterized in that：Pass through single-phase electricity die mould inversion Device, produces exchange auxiliary square-wave voltage and acts on main circuit DC side inductance, improve DC side current waveform, so as to reach suppression The purpose of current on line side harmonic wave processed.

Fig. 1 is the main circuit topology of 18 pulse wave rectifier harmonics restraint systems, and wherein part is exactly that 18 pulse wave rectifiers become Depressor.The phase shift system that rectifier transformer is used is extended triangle phase-shifting, and specific topology is as shown in Fig. 2 AO, BO, CO are one Secondary side winding, aA_o、bB_o、cC_oFor Phase shift winding, A_oB_o、B_oC_o、C_oA_oFor basic winding, by design Phase shift winding and substantially about No-load voltage ratio between group realizes phase shift.If the angle for needing phase shift is α, thenWhenWhen, then 20 ° are differed between rectifier transformer secondary side and primary side three-phase voltage.

Fig. 3 is voltage-type single-phase inversion circuit topological structure, because rectifier bridge output current direction is constant all the time, then single-phase Inverter circuit is only operated in two quadrants, therefore only needs to two full-controlled devices and two diode compositions.

The harmonic suppressing method of the system is divided into following steps：

(1) exemplified by taking first group of rectifier bridge, as shown in figure 4, as 30 ° of 0 ＜ ω t ＜, now diode D₅And D₆Conducting, Then now the voltage expression in Rec1 loops is

As 60 ° of 30 ° of ＜ ω t ＜, now diode D₁And D₆Conducting, now the voltage expression in Rec1 loops is

According to above-mentioned two formula, in order that the voltage on DC inductance is ac square-wave voltage, then need by single-phase inversion electricity Road is by U₁=u_i+ Δ U voltage-drop loading is on main circuit, therefore the output voltage of single-phase inversion circuit is U₁=u_i+ΔU。

(2) DC side load current is I_L, to make 3 groups of diode rectifier bridge load-sharing electric currents and diode rectifier bridge output Electric current linearly rises and linear decline, as shown in figure 5, the amplitude of an appropriate ac square-wave voltage need to be designed.Take first group Exemplified by rectifier bridge, as 30 ° of 0 ＜ ω t ＜, L₁dI_rec1/ dt=-U_S, then electric current I_rec1It is linear to reduce；As 30 ° of 60 ° of ＜ ω t ＜ When, L₁dI_rec1/ dt=U_S, then electric current I_rec1It is linear to rise, as shown in figure 5, then electric current I_rec1Average value beMeet 3 The purpose of individual diode rectifier bridge load-sharing electric current.WhereinClose Reason selection DC inductance L₁Value, the amplitude of ac square-wave voltage is can be obtained by according to above-mentioned formula.

(3) according to the current relationship between rectifier bridge AC and DC side and rectifier transformer primary side and secondary side electricity Relation between stream, passes through rectifier bridge outlet side current waveform, it can be deduced that rectifier transformer primary side current waveform such as Fig. 6 institutes Show, it can be seen that now rectifier transformer primary side A phase current waveforms are sufficiently close to sine wave, and harmonic suppression effect is good.i_a =i '_a1+i′_a2+i′_a3, by calculating, draw：

18 pulse wave rectifier harmonics restraint system emulation conditions：F=50Hz, V=220V, L₁=L₂=L₃=1mH, C=4700 μ F, R =20 Ω, it is assumed that primary side is identical with secondary side output line voltage, definition Then k₁=0.2, k₂=0.684.Full load output current I_L=16.45A, U_o=339V.It is 18 pulse wave rectifier harmonic waves shown in Fig. 7 The current waveform of three first windings of suppression system, it is similar to fictitious current waveform, therefore with the influence of good harmonic carcellation Effect.It is the harmonious wave analysis of 18 pulse electric net side current waveforms for adding single-phase inversion circuit, total harmonic wave shown in Fig. 8 and Fig. 9 Aberration rate is only 1.42%, demonstrate well the present invention in the pulse wave rectifier system net side current harmonics of direct current lateral inhibition 18 just True property and validity.

Claims (1)

1. a kind of method of 18 pulse wave rectifier system harmonicses suppression system, including：Harmonics restraint system；

Described harmonics restraint system includes：18 pulse wave rectifier transformers, 3 three-phase diode uncontrollable rectifier bridges, 3 high frequencies become Depressor, 3 DC inductances, 3 groups of voltage-type single-phase inversion circuits, voltage sensor A, voltage sensor B, voltage sensor C, electricity Pressure sensor D, current sensor E, control circuit and drive circuit；18 pulse wave rectifier transformer inputs get access to grid, 18 pulse waves 3 groups of output ends of rectifier transformer are connected with 3 three-phase diode uncontrollable rectifier bridge inputs respectively, and 3 three-phase diodes are not Control the output end of rectifier bridge and couple load, 3 DC inductances are connected to the output end of three-phase diode uncontrollable rectifier bridge； Voltage-type single-phase inversion circuit direct side is in parallel with system load, and output end is coupled on main circuit by high frequency transformer；Electricity Pressure sensor A, voltage sensor B and voltage sensor C sample the output voltages of 3 three-phase diode rectifier bridges, voltage respectively Sensor D and current sensor E sampling systems load side voltage and electric current, voltage sensor A, voltage sensor B, voltage sensor Device C, voltage sensor D and the result of current sensor E samplings are input to control circuit input end, and control circuit output end is with driving The input of dynamic circuit is connected, and drive circuit exports switching device of the pwm pulse to single-phase inversion circuit；

It is characterized in that：Harmonic suppressing method：Using auxiliary circuit voltage-type single-phase inverter, output produces ac square-wave voltage, By transformer coupled on main circuit DC inductance so that rectifier bridge output current linear change, improve rectifier bridge output electricity Waveform is flowed, suppresses rectifier transformer current on line side harmonic wave；18 pulse wave rectifier system dc side harmonic suppression techniques include following step Suddenly：

(1) AC compounent Δ U acquisition：By taking first group of rectifier bridge as an example, diode rectifier bridge output voltage U_rec1For 6 wave heads DC voltage, therefore its AC compounent Δ U=U_rec1-U_o, U in formula_oDC voltage value is exported for rectifier bridge；

(2) ac square-wave voltage u_iAcquisition：In order to ensure diode output current linear rise and decline, and three-phase diode Rectification circuit energy load-sharing electric current, by taking first group of rectifier bridge as an example, as 30 ° of 0 ＜ ω t ＜, square-wave voltage u_iAmplitude U_S =-L₁di_rec1/ dt, commutation in order to eliminate diode rectifier bridge is delayed and load-sharing electric current, it is necessary to ω t=30 ° when Time forces i_rec1For 0, i in formula_rec1For rectifier bridge output current value；And i_rec1Average value beThen：

$U_S=6{\mathrm{\ωL}}_1\frac{2}{3}{I}_L/\mathrm{\π}=8{\mathrm{fL}}_1{I}_L$

Wherein f is 50Hz, L₁For REC1 DC side inductance, I_LFor load current；Described REC1 is diode uncontrollable rectifier bridge；

(3) auxiliary circuit modulating waveAcquisition：By main circuit AC compounent Δ U^*With ac square-wave voltageIt is added as tune Ripple processed, i.e.,

(4) auxiliary circuit output voltage U₁Acquisition：Auxiliary circuit switch device drive signal is auxiliary circuit modulating waveWith High frequency carrier, which compares, to be obtained, and auxiliary circuit DC side is in parallel with main circuit load-side, obtains output voltage U₁=Δ U+u_i；

(5) rectifier transformer net side A phase currents I_aAcquisition：3 diode rectifications are drawn according to the principle of 18 pulse wave rectifier systems 20 ° are differed in bridge output current wave phase, by calculating, rectifier transformer net side A phase currents I is calculated_aExpression formula such as Under,

$\begin{array}{c}{I}_a=\underset{n=1}{\overset{\mathrm{\∞}}{\mathrm{\Σ}}}\{\frac{32I_L\sin \frac{1}{18}\mathrm{\π}}{{\mathrm{\π}}^2{n}^2}\sin \frac{n\mathrm{\π}}{2}\[2\cos \frac{7}{18}n\mathrm{\π}-3\cos \frac{4}{18}n\mathrm{\π}\\ +4\cos \frac{1}{18}n\mathrm{\π}-3\cos \frac{2}{18}n\mathrm{\π}+2\cos \frac{5}{18}n\mathrm{\π})\·2\cos \frac{1}{18}\mathrm{\π}\\ \·\frac{1}{\sqrt{3}}-\cos \frac{4}{18}n\mathrm{\π}+2\cos \frac{1}{18}n\mathrm{\π}-2\cos \frac{2}{18}n\mathrm{\π}+2\cos \frac{5}{18}n\mathrm{\π}\\ -\cos \frac{8}{18}n\mathrm{\π})\]+\frac{16I_L}{n^2{\mathrm{\π}}^2}\sin \frac{n\mathrm{\π}}{2}(2\cos \frac{1}{6}n\mathrm{\π}-\cos \frac{1}{3}n\mathrm{\π}-1)\}\end{array}$

Inhibit the integral multiple that n in side harmonics, formula is fundamental frequency.