The voltage space phasor representation is a method that directly combines the SPWM and the circular trajectory of the motor flux linkage, which was proposed by J.Holtz in 1983.

According to the principles of electrical engineering, the sine quantity can be represented by a complex number, that is, it can be represented by the projection of a rotating phasor on the imaginary axis on the complex plane.

But the rotating phasor Aejωt on the complex plane [A is called complex amplitude in electrical engineering, A=aejφ0, where a is the absolute value of the complex number A, Aejωt=aejφ0ejωt=aej(φ0+ωt)=acos(ωt+φ0) +jasin(ωt+φ0)] and general space phasors, such as the phasors describing force in mechanics, have different meanings. In order to avoid confusion, the complex amplitude A=aejφ0 corresponding to a sine function is called a space phasor in electrotechnics, and is used to represent it instead. Using complex numbers to represent sine quantities can transform the differential and integral calculations of sinusoidal circuits into algebraic calculations, which greatly simplifies the analysis and calculations of sinusoidal circuits.

A commonly used basic three-phase half-bridge inverter is shown in Figure 1. Among them, Sap~Scn are switch tubes; uA, uB, uC are the voltages added to the load; 0′ is the imaginary midpoint of the load; 0 is the grounding point.

For a three-phase symmetrical load, the following equation holds:

If the A-phase voltage is used as the reference axis (Re axis), the following formula can be used to define the voltage space phasor when using complex coordinates, namely

In the formula, a is the unit phasor operator, a=e2π/3=-1/2+j√3/2. The meaning of multiplying a is to rotate the phasor counterclockwise by 2π/3=120°.

because

so

Equation (4) shows that the space phasor of the voltage on the load is equal to the space phasor of the inverter output to the 0′ point.

In order to study the relationship between the switching mode of the three-phase inverter and the voltage space phasor more conveniently, the following regulations are specially made: each bridge arm of the three-phase inverter has two working modes. On (San off) and San on (Sap off) two working modes. When the Sap is turned on and the San is turned off, the working mode of the A phase is represented by 1 (that is, the voltage of the A point to the 0′ point is +E/2, and the negative pole of the DC voltage is E); when the San is turned on and the Sap is turned off When it is off, the working mode of phase A is represented by 0 (that is, the voltage of point A to 0′ point is -E/2, and the negative pole of the DC voltage is 0). Another example is the B-phase bridge arm, when Sbp is turned on and Sbn is turned off, the working mode of B-phase is represented by 1 (the voltage of B point to 0′ point is E/2, and the negative pole of DC voltage is E); when Sbn When it is turned on and Sbp is turned off, the working mode of the B phase is represented by 0 (the voltage of the B point to the 0′ point is -E/2, and the negative electrode of the DC voltage is 0). The same is true for the C-phase bridge arm. When Scp is turned on, the C-phase operating mode is represented by 1; when Scn is turned on, the C-phase operating mode is represented by 0. Since each phase bridge arm has two working modes, there are 3 phases in total, so there are 23=8 working modes for the 6 switching tubes Sap~Scn of the three-phase half-bridge inverter.

The 8 working modes of the three bridge arms are represented by three digits in the order of A, B, and C, denoted as (000), (111), (100), (001), (110), (011), (101), (010). (000) means that the voltages of the three points A, B, and C to the 0′ point are all equal to -E/2; (111) means that the voltages of the three points A, B, and C to the 0′ point are all equal to E/2; (100) It means that the voltage of point A to point O’ is E/2, and the voltage of point B and C to point 0′ is -E/2; (110) means that the voltage of point A and point B to point 0′ is E/2, and the voltage of point C to point 0′ is E/2. The voltage from point to point 0′ is -E/2. …

Let’s study the voltage space phasor of the three-phase half-bridge SPWM inverter with a switching period of 180° (as shown in Figure 2-57), as well as the waveform diagram of this inverter and the operation of the six switching tubes. model. According to the switching sequence of Sap~Scn, a cycle is divided into 6 parts, each part has a pair of switches for conversion, so there are 6 switching modes, as shown in the lower part of Figure 2-57, that is, Sap, Sbp, Scn are turned on. (100); Sap, Sbp, Scn. On (110); San, Sbp, Scn on (010); San, Sbp, Scp on (011); San, Sbn, Scp on (001); Sap, Sbn, Scp on of (101). The voltage space phasors of these six switching modes are used to represent them in turn.

When the A-phase voltage is used as the Re axis of the reference phase, the space phasor of the voltage is

The value of the output voltage ū_{AO’}, ū_{BO’}, ū_{OO’} of each phase of the inverter has been explained before, and they are +E/2 or -E/2 respectively, so the known voltage space phasors are

The voltage space phasors (ie complex coordinate voltage phasors) of Ū0 and ū1-ū7 are shown in Figure 3(a). where ū0(000) and ū7(111) are at the origin of complex plane coordinates.

The integral of the instantaneous voltage space phasor over time is the trajectory of the space flux linkage phasor, that is, ψ=∫ū*Δt, the flux linkage increment Δψ=∫ū*dt=ū·ΔT, Δψk=Eej ( k-1)π/3`△T, k=1,2,3,…6.

Assuming that the load of the three-phase inverter is an asynchronous motor, if the voltage drop on the stator winding is ignored, it is the trajectory of the magnetic flux linkage phasor in the stator space. Since the complex amplitude of each non-zero voltage space phasor is E, we have

In the formula, ψ0 is the initial space flux linkage phasor; E·ΔT=Δψ is called the magnetic flux linkage increment.

Figure 3(b) is the regular hexagonal flux linkage track formed by the output voltage phasor of the 180° conduction type three-phase inverter.

In order to adjust the output voltage and frequency of the inverter, ū0△T or ū7△T can be introduced on the flux linkage track. Its function is to stop the magnetic flux linkage track, that is, during the ΔT time, the magnetic flux linkage phasor stays in place, and the output voltage and frequency can be adjusted by adjusting the size of ΔT.

The space phasor of the output voltage of the three-phase half-bridge SPWM inverter is studied below. Taking the three-phase half-bridge SPWM inverter with carrier ratio F=6 as an example, its working waveform and switching mode are shown in Figure 4. The half cycle of the output line voltage consists of 6 square pulses, and one cycle consists of 12 pulses. The analysis shows that the positive pulse of line voltage u_{AB} is composed of voltage phasors of two switching modes, ū1 (100) and ū6 (101); the negative pulse of u_{AB} is composed of two switching modes of ū3 (010) and ū4 (011). It consists of voltage phasors; it consists of zero pulses of u_{AB}. Ū0 (000), ū7 (111), ū5 (001) and ū2 (110) 4 switching modes of voltage phasor composition. The distribution area of the line voltage “u_{AB}, u_{BC}, u_{CA} in the output voltage space phasor diagram is shown in Figure 5. The values of Ū0 (000) and ū7 (111) are equal to zero and are located at the origin of the complex plane coordinates; the phasor ū1 The value of ~ū6 is equal to E, starting from the Re axis on the complex plane coordinates, and uniformly distributed at 60° to each other.

For the three-phase half-bridge SPWM inverter, assuming that the load is an asynchronous motor, when the voltage drop on the stator winding of the asynchronous motor is ignored, the trajectory of the spatial flux linkage phasor ψ can be obtained by integrating the voltage phasor . For a three-phase half-bridge SPWM inverter with a carrier ratio of F, since its phase voltage is no longer a 180° square wave, but a SPWM waveform composed of F positive and negative pulses alternately, its space voltage phasor is In one cycle, it consists of 6F phasors from ū0 to ū7. This shows that the spatial phasor ū of the output voltage of the SPWM three-phase half-bridge inverter is composed of 6F pulses with an amplitude of 0 or E/2, and it changes, so ψ=∫ū(t)dt The trajectory of is a circle consisting of 6F polylines. When F approaches ∞, the trajectory of ψ tends to be a circle; when F=1, the trajectory of ψ is a regular hexagon.

The above is the space phasor representation of the three-phase half-bridge SPWM inverter. With this representation, the modulation methods of some SPWM three-phase half-bridge inverters can be obtained according to the space phasor. Especially for the three-phase half-bridge inverter for AC drive, it is convenient and intuitive to use the space phasor method to analyze.

The space phasor SPWM modulation method has gone beyond the conventional SPWM idea. From the perspective of the motor, it directly aims to control the circular trajectory of the motor magnetic flux chain. It not only has the same effect as SPWM in control, but also is more intuitive, the physical meaning is clearer, and it is more convenient to implement. More importantly, the space phasor SPWM modulation method is superior to the SPWM method in terms of DC voltage utilization and motor harmonic loss, and its maximum modulation degree M=0.907>0.866. The reason is that the potential at the 0′ point floats according to the 3rd harmonic, and the same 3rd harmonic is injected into each phase voltage, which reduces the peak value of the combined phase voltage to the midpoint O of the DC power supply voltage.

In the space phasor modulation, the zero phasor action time is zero, which corresponds to the maximum modulation degree M, which corresponds to the circular trajectory formed by the reference phasor ū* on the phasor diagram and the regular hexagonal phase formed by the switching phasor. The moment of cutting, as shown in Figure 6. This also means that the conventional SPWM modulation method has reached the modulation limit. At this time, if you want to continue to increase the modulation degree M, you must use the overmodulation technique. This must be noted.

Overmodulation has two regions, corresponding to two modes, namely mode 1 and mode 2. The area from the inscribed circle of the regular hexagon to the regular hexagon is mode 1, and when it reaches the regular hexagon, it is mode 2, as shown in Figure 6(b).

Mode 1: The trajectory of the reference phasor ū* consists of the arc in the regular hexagon and the middle part of each side of the regular hexagon. When * moves on the arc, the action time of each phasor is obtained by the SVM method; when ū** moves on each side of the regular hexagon, the action time of the zero phasors ū0 and ū7 is 0, and The action time of the non-zero phasors ū1~ū6 is

In the formula, T8 is the modulation wave period; φ is the phase angle.

Mode 2: When it is necessary to continue to increase the modulation degree to M=0.952, the arc length in the regular hexagon is reduced to zero. The trajectory of Ū* becomes a pure hexagon and enters the mode 2 area. At this time, it is overmodulated according to mode 2. In this mode, the speed of the voltage phasor is changed by adjusting the action time of two adjacent working phasors. In the middle part of each side of the regular hexagon, the movement speed is faster, and the speed is slower on the two sides. In this way, a smooth transition from mode 2 to six-step wave operation is possible. Compared with the waveform obtained by forced overmodulation, the distortion rate of the waveform obtained by this overmodulation technique is much smaller.