Figure 1. shows the typical structure of a three-phase induction motor connected to a VSI (Voltage Source Inverter). Since the motor is considered as a balanced load with an unconnected neutral ,Vn=(Va+Vb+Vc)/3 , Van=Va-Vn=(Vab-Vac) /3 , Vbn=Vb-Vn=(Vbc-Vac) /3 and Vcn=Vc-Vn=(Vca-Vbc) /3 . Since the upper power switches can only be On or Off, and since the lower ones are supposed to always be in the opposed state (the dead-times of the inverter legs are neglected), there are only eight possible switching states, as shown on Figure 2. Six of voltages.them lead to non-zero phase voltages, and two interchangeable states lead to zero phase When mapped in a 2D-frame fixed to the stator using a Concordia transformation, the six non-zero phase voltages form the vertices of a hexagon. (See Figure 3.)
As shown on Figure 3. , the angle between two successive non-zero voltages is always 60 degrees.
In complex form, these non-zero phase voltages can be written as Vk , with k = 1..6 and Vo-v7=0V. Table 1. shows the line-to-line and line-to-neutral voltages in each of the 8 possible configurations of the inverter.
In the Concordia frame, any stator voltage Vs = Vα + jVβ =Vsm cos(θ) + jVsm sin(θ) located insidethis hexagon belongs to one of the six sectors, and can be expressed as a linear combination of the two non-zero phase voltages which delimit this sector: Vs=dkVk+(dk+1)(Vk+1). Equating dkVk+(dk+1)(Vk+1) to Vsm cos(θ) + jVsm sin(θ) in each sector leads to the expressions of the dutycycles shown in Table 2. Since the inverter cannot instantaneously generate Vs the spacevector PWM principle consists in producing a Ts-periodic voltage whose average value equal Vs by generating Vk during Tk=dkTs and Vk+1 during Tk+1 = (dk+1)Ts.
In the Concordia frame, any stator voltage Vs = Vα + jVβ =Vsm cos(θ) + jVsm sin(θ) located insidethis hexagon belongs to one of the six sectors, and can be expressed as a linear combination of the two non-zero phase voltages which delimit this sector: Vs=dkVk+(dk+1)(Vk+1). Equating dkVk+(dk+1)(Vk+1) to Vsm cos(θ) + jVsm sin(θ) in each sector leads to the expressions of the dutycycles shown in Table 2. Since the inverter cannot instantaneously generate Vs the spacevector PWM principle consists in producing a Ts-periodic voltage whose average value equal Vs by generating Vk during Tk=dkTs and Vk+1 during Tk+1 = (dk+1)Ts.
Since dk+(dk+1) these voltages must be completed over the switching period To by Vo and/or V7 . Several solutions are possible , and the one which minimizes the total harmonic distorsion of the stator current consists i applying Vo and V7 during the same duration To=T7 is equally applied at the beginning and at the end of the switching period, whereas is applied at the middle. As an illustration, the upper side of
Figure 4 . shows the waveforms obtained in sector 1
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