CN103560528A - STATCOM control system and method based on improved instantaneous reactive power - Google Patents

Abstract

Provided is an STATCOM control system and method based on improved instantaneous reactive power. The system is characterized by comprising an acquisition circuit, a control circuit, a drive circuit, a main circuit and a protective circuit. The method includes the steps of data collecting, processing, controlling and driving. The system and method have the advantages of being simple in design, low in cost, high in operation speed, capable of increasing real-time response speed of an STATCOM and improving stability of a power system, simple and understandable in detection method, easy to program and beneficial for being widely popularized.

Description

STATCOM control system and method based on improving instantaneous reactive power

(1) technical field:

The invention belongs to electric power system STATCOM (StaticSynchronous Compensator---STATCOM) Test and control field, especially a kind of STATCOM control system and method based on improving instantaneous reactive power.

(2) background technology:

STATCOM (STATCOM) is the important component part that belongs to flexible flexible AC transmitting system (FACTS), is the representative of current reactive power compensation field state-of-the-art technology.Along with the development increasingly of device for high-power power electronic, the application of STATCOM in electric power system is also more and more extensive.In power distribution network, STATCOM can effectively solve the multiple power quality problems such as voltage fluctuation, voltage dip, Voltage unbalance, harmonic pollution.Therefore become one of focus of current electrical engineering area research.

The basic principle of STATCOM is to utilize can turn-off device for high-power power electronic (as IGBT, GTO etc.) form from commutation bridge circuit, through reactor, be connected in parallel on electrical network, suitably regulate amplitude and the phase place of bridge circuit AC output voltage, or directly control its ac-side current, just can make this circuit absorb or send the reactive current meeting the demands, realize the object of dynamic passive compensation.Can be accurately and detect fast the key that reactive current is reactive power compensation technology.So it is particularly important that the research of the accurate detection method of STATCOM seems.The instantaneous reactive power detection method of extensive use at present can only can detect exactly the reactive current in system in desirable line voltage situation.Its reason is when line voltage is asymmetric, and actual zero crossing and the voltage fundamental zero crossing of voltage have phase difference.And in the actual a lot of situations of grid voltage waveform, all there is distortion in various degree and asymmetric.

(3) summary of the invention:

The object of the invention is to provide a kind of STATCOM control system and method based on improving instantaneous reactive power, it can overcome the deficiency of existing instantaneous reactive power detection algorithm, by dq, convert the fundamental positive sequence that synchronously extracts voltage and current, through coordinate transform, thereby phase shift draws current first harmonics positive sequence idle component, can be real-time detect three-phase voltage distortion and fundamental positive sequence reactive current when asymmetric, be the control system that a kind of computational accuracy is high, detection accuracy is high, method is simple.

Technical scheme of the present invention: a kind of STATCOM control system based on improving instantaneous reactive power, is characterized in that it comprises Acquisition Circuit, control circuit, drive circuit, main circuit and protective circuit; Wherein, the input of described Acquisition Circuit connects the voltage transformer in external electrical network, and its output connects the input of control circuit; The output of described control circuit connects the input on drive circuit; The output of described drive circuit connects the input of main circuit; The output output compensating current signal of described main circuit feeds back to electrical network; Described protective circuit input connects DC bus current instrument transformer, and its output connects the input of control circuit.

Described Acquisition Circuit is comprised of voltage transformer, diode D11, diode D12, capacitor C 11, resistance R 11, amplifying circuit OP07, resistance R 21, resistance R 22, resistance R 23, resistance R 24, diode D21, diode D22; After the signal of collection is processed, send into control circuit analog-to-digital conversion module port, it is connected to conventional connection.

Described voltage transformer adopts TA10104-2K horizontal piercing Miniature precision AC voltage transformer.

Described control circuit is comprised of analog-to-digital conversion module, dsp chip, data memory module; Described digital-to-analogue dress changes module input and is connected with Acquisition Circuit output, and its output is two-way with dsp chip input and is connected; Described DSP input is connected with protective circuit output simultaneously, and its output is connected with drive circuit input, is two-way is connected with data memory module simultaneously.

Described analog-to-digital conversion module adopts the ADC conversion chip ADS8364 of TI company.

Described dsp chip adopts TMS320F2812PDSP chip, and it has RAM on the sheet of 18K*16bit, 128K in-chip FLASH, memory on 128K plate, with host computer by RS232 interface communication.

Described drive circuit is by driving chip, resistance R 31, resistance R 32, resistance R 33, resistance R 34, resistance R 35, resistance R 36, capacitor C 31, capacitor C 32, diode D31, voltage-stabiliser tube DW1, voltage-stabiliser tube DW2 to form; To drive signal input main circuit to trigger IGBT; The mixing IC driver EXB841 that described driving chip adopts company of Fuji to produce.

Described main circuit consists of three-phase inversion bridge-type circuit and capacitor C; Described each brachium pontis all has an IGBT switching tube and a diode inverse parallel to form, and one end of each brachium pontis connects respectively the three-phase line of electrical network, and the other end connects capacitor C; The gate pole of described each IGBT switching tube is accepted the driving signal that drive circuit sends.

The IGBT module that described IGBT adopts EUPEC company to produce, model is BSM50GB120DN.

Described protective circuit is comprised of current sensor, LM311 comparator, resistance R 41, resistance R 42 resistance R 43, resistance R 44, capacitor C 41, capacitor C 42.Detect the electric current on inverter bridge input direct-current bus, output signal is sent into fracture in the INT of control circuit DSP; The direct measuring type hall effect current sensor HAPS 1 that described current sensor adopts Japanese HINooE company to produce.

STATCOM measurement and control system and a method of improving instantaneous reactive power, is characterized in that comprising the following steps:

(1) analog-to-digital conversion module that Acquisition Circuit is detected to the voltage signal input control circuit obtaining is changed.

(2) by the dsp chip of the digital signal input control circuit of analog-to-digital conversion module output.

(3) digital signal that dsp chip utilization obtains is carried out the calculating of reactive current, obtains the size of required compensation rate after adjusting, then compensation rate is converted into modulation signal, then calculates the SPWM pulse that response is sent in SPWM pulse, sends into drive circuit.

(4) drive circuit sends the break-make of 6 IGBT switching devices in signal controlling main circuit.

(5) break-make of each brachium pontis of signal deciding that each brachium pontis basis in main circuit receives is to produce the reactive current of expected compensation.

Detecting Reactive Current in described step (3) is calculated and is consisted of following steps:

1. voltage signal Acquisition Circuit being obtained carries out coordinate transform by the dsp chip of control circuit;

2. the signal that passes through changes in coordinates carries out digital low-pass filtering and obtains its DC component in dsp chip, and DC component calculates the initial phase angle θ of voltage fundamental positive sequence component in real time by dsp chip;

$\mathrm{\&theta;}=\left\{\begin{array}{cc}{\mathrm{tg}}^{-1}\left[\frac{{e_d}^{-}}{-{e_q}^{-}}\right]& ({e_d}^{-}\&GreaterEqual;0)\\ \mathrm{\&pi;}+{\mathrm{tg}}^{-1}\left[\frac{{e_d}^{-}}{-{e_q}^{-}}\right]& ({e_d}^{-}<0)\end{array}\right.$

3. dsp chip carries out phase shift to the voltage signal obtaining in Acquisition Circuit, wherein phase shift is by the sinusoidal signal sin/cos in figure (4), carry out phase place correction, correction angle is θ, and the voltage signal phase place after phase shift is sin (ω t+ θ) and cos (ω t+ θ).

4. the mains voltage signal after the phase shift obtaining in 3. according to step, in dsp chip, according to coordinate transform formula T, power network current signal is carried out to coordinate transform, through the signal continuation of changes in coordinates, in dsp chip, carry out digital low-pass filtering simultaneously and obtain its DC component, then pass through coordinate inverse transformation formula T ^-1obtain the command signal of reactive current.

Wherein,

$T=\sqrt{\frac{2}{3}}g\left[\begin{array}{ccc}\sin (\mathrm{\&omega;t}+\mathrm{\&theta;})& \sin (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& \sin (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\\ -\cos (\mathrm{\&omega;t}+\mathrm{\&theta;})& \text{-cos}(\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& -\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\end{array}\right]$

$T^{-1}=\sqrt{\frac{2}{3}}g\left[\begin{array}{cc}\sin (\mathrm{\&omega;t}+\mathrm{\&theta;})& \text{-cos}(\mathrm{\&omega;t}+\mathrm{\&theta;})\\ \sin (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& -\cos (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\\ \sin (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& -\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\end{array}\right].$

Operation principle of the present invention:

STATCOM Detecting Reactive Current algorithm based on improving instantaneous reactive power theory, first, by 3/2 conversion, is transformed into three-phase current to be detected, voltage α β coordinate system from abc coordinate system, calculates transient current i _pand i _q, then through low pass filter LPF, obtain its DC component under α β coordinate system

with

, the humorous negative sequence component that involves becomes alternating current component by LPF filtering at α β coordinate system.In like manner, we can obtain with , corresponding to voltage e _a, e _b, e _cfundamental positive sequence.Above-mentioned a series of computing is similar with traditional reactive current detection method based on instantaneous reactive power theory.

By i _pand i _qresultant current vector i project to e _pand e _qsynthesized voltage vector e upper, the current phasor i' after being adjusted, this vector is corresponding to current i _a, i _band i _cfundamental positive sequence real component.Through inverse operation, obtain the fundamental positive sequence real component i' under abc coordinate system _af, i' _bf, i' _cf.This component and difference between currents to be detected are reactive current component contained in system power.

Superiority of the present invention is: 1, hardware unit combines with computer software programming, simplicity of design, and cost is lower, is easy to realize; 2, adopt the detection method of improving instantaneous reactive power theory, when three-phase voltage occurs to distort and be asymmetric, still can detect exactly the reactive current of system; 3, the acp chip of control circuit adopts DSP, and its execution speed is fast, and from improving the real time response speed of STATCOM, strengthens the stability of electric power system; 4, detection method is easily understood, and easily programming, is conducive to application.

(4) accompanying drawing explanation:

Fig. 1 is the related a kind of general structure schematic diagram based on improving the STATCOM control system of instantaneous reactive power of the present invention.

Fig. 2 is Acquisition Circuit structural representation in the related a kind of STATCOM control system based on improvement instantaneous reactive power of the present invention.

Fig. 3 is control circuit structural representation in the related a kind of STATCOM control system based on improvement instantaneous reactive power of the present invention.

Fig. 4 is driving circuit structure schematic diagram in the related a kind of STATCOM control system based on improvement instantaneous reactive power of the present invention.

Fig. 5 is main circuit structure schematic diagram in the related a kind of STATCOM control system based on improvement instantaneous reactive power of the present invention.

Fig. 6 is protective circuit structural representation in the related a kind of STATCOM control system based on improvement instantaneous reactive power of the present invention.

(5) embodiment:

Embodiment: a kind of STATCOM control system (see figure 1) based on improving instantaneous reactive power, is characterized in that it comprises Acquisition Circuit, control circuit, drive circuit, main circuit and protective circuit; Wherein, the input of described Acquisition Circuit connects the voltage transformer in external electrical network, and its output connects the input of control circuit; The output of described control circuit connects the input on drive circuit; The output of described drive circuit connects the input of main circuit; The output output compensating current signal of described main circuit feeds back to electrical network; Described protective circuit input connects DC bus current instrument transformer, and its output connects the input of control circuit.

Described Acquisition Circuit (see figure 2) is comprised of voltage transformer, diode D11, diode D12, capacitor C 11, resistance R 11, amplifying circuit OP07, resistance R 21, resistance R 22, resistance R 23, resistance R 24, diode D21, diode D22; After the signal of collection is processed, send into control circuit analog-to-digital conversion module port, it is connected to conventional connection.

Described voltage transformer (see figure 2) adopts TA10104-2K horizontal piercing Miniature precision AC voltage transformer.

Described control circuit (see figure 3) is comprised of analog-to-digital conversion module, dsp chip, data memory module; Described digital-to-analogue dress changes module input and is connected with Acquisition Circuit output, and its output is two-way with dsp chip input and is connected; Described DSP input is connected with protective circuit output simultaneously, and its output is connected with drive circuit input, is two-way is connected with data memory module simultaneously.

Described analog-to-digital conversion module adopts the ADC conversion chip ADS8364 of TI company.

Described dsp chip adopts TMS320F2812PDSP chip, and it has RAM on the sheet of 18K*16bit, 128K in-chip FLASH, memory on 128K plate, with host computer by RS232 interface communication.

Described drive circuit (see figure 4) is by driving chip, resistance R 31, resistance R 32, resistance R 33, resistance R 34, resistance R 35, resistance R 36, capacitor C 31, capacitor C 32, diode D31, voltage-stabiliser tube DW1, voltage-stabiliser tube DW2 to form; To drive signal input main circuit to trigger IGBT; The mixing IC driver EXB841 that described driving chip adopts company of Fuji to produce.

Described main circuit (see figure 5) consists of three-phase inversion bridge-type circuit and capacitor C; Described each brachium pontis all has an IGBT switching tube and a diode inverse parallel to form, and one end of each brachium pontis connects respectively the three-phase line of electrical network, and the other end connects capacitor C; The gate pole of described each IGBT switching tube is accepted the driving signal that drive circuit sends.

The IGBT module that described IGBT adopts EUPEC company to produce, model is BSM50GB120DN.

Described protective circuit (see figure 6) is comprised of current sensor, LM311 comparator, resistance R 41, resistance R 42 resistance R 43, resistance R 44, capacitor C 41, capacitor C 42.Detect the electric current on inverter bridge input direct-current bus, output signal is sent into fracture in the INT of control circuit DSP; The direct measuring type hall effect current sensor HAPS 1 that described current sensor adopts Japanese HINooE company to produce.

STATCOM measurement and control system and a method of improving instantaneous reactive power, is characterized in that comprising the following steps:

(1) analog-to-digital conversion module that Acquisition Circuit is detected to the voltage signal input control circuit obtaining is changed.

(2) by the dsp chip of the digital signal input control circuit of analog-to-digital conversion module output.

(3) digital signal that dsp chip utilization obtains is carried out the calculating of reactive current, obtains the size of required compensation rate after adjusting, then compensation rate is converted into modulation signal, then calculates the SPWM pulse that response is sent in SPWM pulse, sends into drive circuit.

(4) drive circuit sends the break-make of 6 IGBT switching devices in signal controlling main circuit.

(5) break-make of each brachium pontis of signal deciding that each brachium pontis basis in main circuit receives is to produce the reactive current of expected compensation.

Detecting Reactive Current in described step (3) is calculated and is consisted of following steps:

1. voltage signal Acquisition Circuit being obtained carries out coordinate transform by the dsp chip of control circuit;

2. the signal that passes through changes in coordinates carries out digital low-pass filtering and obtains its DC component in dsp chip, and DC component calculates the initial phase angle θ of voltage fundamental positive sequence component in real time by dsp chip;

$\mathrm{\&theta;}=\left\{\begin{array}{cc}{\mathrm{tg}}^{-1}\left[\frac{{e_d}^{-}}{-{e_q}^{-}}\right]& ({e_d}^{-}\&GreaterEqual;0)\\ \mathrm{\&pi;}+{\mathrm{tg}}^{-1}\left[\frac{{e_d}^{-}}{-{e_q}^{-}}\right]& ({e_d}^{-}<0)\end{array}\right.$

3. dsp chip carries out phase shift to the voltage signal obtaining in Acquisition Circuit, wherein phase shift is by the sinusoidal signal sin/cos in figure (4), carry out phase place correction, correction angle is θ, and the voltage signal phase place after phase shift is sin (ω t+ θ) and cos (ω t+ θ).

4. the mains voltage signal after the phase shift obtaining in 3. according to step, in dsp chip, according to coordinate transform formula T, power network current signal is carried out to coordinate transform, through the signal continuation of changes in coordinates, in dsp chip, carry out digital low-pass filtering simultaneously and obtain its DC component, then pass through coordinate inverse transformation formula T ^-1obtain the command signal of reactive current.

Wherein,

$T=\sqrt{\frac{2}{3}}g\left[\begin{array}{ccc}\sin (\mathrm{\&omega;t}+\mathrm{\&theta;})& \sin (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& \sin (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\\ -\cos (\mathrm{\&omega;t}+\mathrm{\&theta;})& \text{-cos}(\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& -\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\end{array}\right]$

$T^{-1}=\sqrt{\frac{2}{3}}g\left[\begin{array}{cc}\sin (\mathrm{\&omega;t}+\mathrm{\&theta;})& \text{-cos}(\mathrm{\&omega;t}+\mathrm{\&theta;})\\ \sin (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& -\cos (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\\ \sin (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& -\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\end{array}\right].$

Claims (10)

1. the STATCOM control system based on improving instantaneous reactive power, is characterized in that it comprises Acquisition Circuit, control circuit, drive circuit, main circuit and protective circuit; Wherein, the input of described Acquisition Circuit connects the voltage transformer in external electrical network, and its output connects the input of control circuit; The output of described control circuit connects the input on drive circuit; The output of described drive circuit connects the input of main circuit; The output output compensating current signal of described main circuit feeds back to electrical network; Described protective circuit input connects DC bus current instrument transformer, and its output connects the input of control circuit.

2. a kind of STATCOM control system based on improving instantaneous reactive power according to claim 1, is characterized in that described Acquisition Circuit is comprised of voltage transformer, diode D11, diode D12, capacitor C 11, resistance R 11, amplifying circuit OP07, resistance R 21, resistance R 22, resistance R 23, resistance R 24, diode D21, diode D22; After the signal of collection is processed, send into control circuit analog-to-digital conversion module port, it is connected to conventional connection.

3. a kind of STATCOM control system based on improving instantaneous reactive power according to claim 2, is characterized in that described voltage transformer adopts TA10104-2K horizontal piercing Miniature precision AC voltage transformer.

4. a kind of STATCOM control system based on improving instantaneous reactive power according to claim 1, is characterized in that described control circuit is comprised of analog-to-digital conversion module, dsp chip, data memory module; Described digital-to-analogue dress changes module input and is connected with Acquisition Circuit output, and its output is two-way with dsp chip input and is connected; Described DSP input is connected with protective circuit output simultaneously, and its output is connected with drive circuit input, is two-way is connected with data memory module simultaneously.

5. a kind of STATCOM control system based on improving instantaneous reactive power according to claim 4, is characterized in that described analog-to-digital conversion module adopts the ADC conversion chip ADS8364 of TI company; Described dsp chip adopts TMS320F2812PDSP chip, and it has RAM on the sheet of 18K*16bit, 128K in-chip FLASH, memory on 128K plate, with host computer by RS232 interface communication.

6. a kind of STATCOM control system based on improving instantaneous reactive power according to claim 1, is characterized in that described drive circuit is by driving chip, resistance R 31, resistance R 32, resistance R 33, resistance R 34, resistance R 35, resistance R 36, capacitor C 31, capacitor C 32, diode D31, voltage-stabiliser tube DW1, voltage-stabiliser tube DW2 to form; To drive signal input main circuit to trigger IGBT; The mixing IC driver EXB841 that described driving chip adopts company of Fuji to produce.

7. a kind of STATCOM control system based on improving instantaneous reactive power according to claim 1, is characterized in that described main circuit consists of three-phase inversion bridge-type circuit and capacitor C; Described each brachium pontis all has an IGBT switching tube and a diode inverse parallel to form, and one end of each brachium pontis connects respectively the three-phase line of electrical network, and the other end connects capacitor C; The gate pole of described each IGBT switching tube is accepted the driving signal that drive circuit sends; The IGBT module that described IGBT adopts EUPEC company to produce, model is BSM50GB120DN.

8. a kind of STATCOM control system based on improving instantaneous reactive power according to claim 1, is characterized in that described protective circuit is comprised of current sensor, LM311 comparator, resistance R 41, resistance R 42 resistance R 43, resistance R 44, capacitor C 41, capacitor C 42.Detect the electric current on inverter bridge input direct-current bus, output signal is sent into fracture in the INT of control circuit DSP; The direct measuring type hall effect current sensor HAPS 1 that described current sensor adopts Japanese HINooE company to produce.

9. STATCOM measurement and control system and the method based on improving instantaneous reactive power, is characterized in that comprising the following steps:

(1) analog-to-digital conversion module that Acquisition Circuit is detected to the voltage signal input control circuit obtaining is changed.

(2) by the dsp chip of the digital signal input control circuit of analog-to-digital conversion module output.

(3) digital signal that dsp chip utilization obtains is carried out the calculating of reactive current, obtains the size of required compensation rate after adjusting, then compensation rate is converted into modulation signal, then calculates the SPWM pulse that response is sent in SPWM pulse, sends into drive circuit.

(4) drive circuit sends the break-make of 6 IGBT switching devices in signal controlling main circuit.

(5) break-make of each brachium pontis of signal deciding that each brachium pontis basis in main circuit receives is to produce the reactive current of expected compensation.

10. a kind of STATCOM measurement and control system and method based on improving instantaneous reactive power according to claim 9, is characterized in that Detecting Reactive Current in described step (3) is calculated to consist of following steps:

1. voltage signal Acquisition Circuit being obtained carries out coordinate transform by the dsp chip of control circuit;

2. the signal that passes through changes in coordinates carries out digital low-pass filtering and obtains its DC component in dsp chip, and DC component calculates the initial phase angle θ of voltage fundamental positive sequence component in real time by dsp chip;

$\mathrm{\&theta;}=\left\{\begin{array}{cc}{\mathrm{tg}}^{-1}\left[\frac{{e_d}^{-}}{-{e_q}^{-}}\right]& ({e_d}^{-}\&GreaterEqual;0)\\ \mathrm{\&pi;}+{\mathrm{tg}}^{-1}\left[\frac{{e_d}^{-}}{-{e_q}^{-}}\right]& ({e_d}^{-}<0)\end{array}\right.$

3. dsp chip carries out phase shift to the voltage signal obtaining in Acquisition Circuit, wherein phase shift is by the sinusoidal signal sin/cos in figure (4), carry out phase place correction, correction angle is θ, and the voltage signal phase place after phase shift is sin (ω t+ θ) and cos (ω t+ θ).

4. the mains voltage signal after the phase shift obtaining in 3. according to step, in dsp chip, according to coordinate transform formula T, power network current signal is carried out to coordinate transform, through the signal continuation of changes in coordinates, in dsp chip, carry out digital low-pass filtering simultaneously and obtain its DC component, then pass through coordinate inverse transformation formula T ^-1obtain the command signal of reactive current.

Wherein,

$T=\sqrt{\frac{2}{3}}g\left[\begin{array}{ccc}\sin (\mathrm{\&omega;t}+\mathrm{\&theta;})& \sin (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& \sin (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\\ -\cos (\mathrm{\&omega;t}+\mathrm{\&theta;})& \text{-cos}(\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& -\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\end{array}\right]$

$T^{-1}=\sqrt{\frac{2}{3}}g\left[\begin{array}{cc}\sin (\mathrm{\&omega;t}+\mathrm{\&theta;})& \text{-cos}(\mathrm{\&omega;t}+\mathrm{\&theta;})\\ \sin (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& -\cos (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\\ \sin (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})& -\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;}}{3}+\mathrm{\&theta;})\end{array}\right].$

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