Determination of the switching sequence

In order to determine the switching sequence, four representative triangles will be examined OCD (inner), CDE (intermediate), ACE and BDE (outers), as shown in Fig. 16(a). Additionally the hexagon is divided in A-B subsectors determined by comparison of corresponding application times of the same inverter, e.g. inverter H:

bH aH t

t ≥ , (14)

The voltage reference vector v* lays in one of these triangles, leading to the following four relevant cases.

A. Inner triangle OCD

For the inner triangle the NTV are v_O, v_C, and v_D which can be obtained by the following combinations of voltage vectors selected for the two inverters:

• (v_oH,v_oL), forming v_O,

• (v_aH,v_oL) and (v_oH,v_aL), forming v_C,

• (v_bH,v_oL) and (v_oH,v_bL), forming v_D.

By using these combinations, since v^*=v_H^* +v_L^* is inside the inner triangle, for any value of v_H^* and v_L^*, Fig. 16(b), the total application time of active vectors is

s

By combining (15) with (2) yields

S oL

oH t T

t + ≥ (16)

Note that (16) provides a criterion for the identification of the inner triangle OCD. The voltage vector combinations can be arranged within the switching period to obtain a switching sequence suitable for the implementation in PWM generation unit of industrial DSPs, as represented in Fig. 17(a). In the figure is emphasized (grey) the overlap between the two null vectors (v_oH and

D E

by using the two adjacent active vectors for each inverter.

voL), provided by (16), which secures the generation of v* only by vectors v_O, v_C, and v_D. For the B-half the order of the duty cycles remains the same (e.g. TaH > TbH > TcH), they just increase/decrease.

The proposed switching sequence belongs to symmetrical and discontinuous modulation, minimizing the number of commutations. A continuous modulation can be easily obtained by introducing the null vector v_O in the middle and at the ends of the switching period.

B. Outer triangle ACE

For the first outer triangle the NTV are v_A, v_C, and v_E, which can be composed by the combinations:

• (v_aH,v_aL), forming v , _A

• (v_aH,v_oL) and (v_oH,v_aL), forming v , _C

• (v_aH,v_bL) and (v_bH,v_aL), forming v . _E

Since v* lies inside the triangle ACE, its component along v_aH is bigger than the amplitude of vaH (see Fig. 16). Similar to the previous case, for application times this consideration leads to

S aL

aH t T

t + ≥ (17)

Equation (17) provides a criterion for the identification of the outer triangle ACE. Also in this case, the voltage vector combinations can be arranged within the switching period to obtain a switching sequence suitable for the implementation in PWM generation unit of industrial DSPs, as represented in Fig. 18(a). In the figure is emphasized the overlap between the two active vectors (v_aH and v_aL), provided by (17), which allows the generation of v* only by vectors v_A,

vC and v_E. As in the previous case of the inner triangle, the proposed sequence leads to symmetrical and discontinuous modulation, with the difference that is not possible anymore to introduce continuous modulation.

v_O Fig. 17. Proposed switching sequences for inner triangle OCD case (a) A-half, (b) B-half.

C. Triangle BDE

For the second outer triangle the NTV are v_B, v_D, and v_E. Due to the symmetry of outer triangles ACE and BDE, this case can be treated as the previous one, involving vectors v_bH and

vbL instead of v_aH and v_aL, respectively, leading to

S bL

bH t T

t + ≥ (18)

Also in this case, (18) provides a criterion for the identification of the outer triangle BDE. The proposed switching sequence is shown in Fig. 18(b), again symmetrical and discontinuous as in the previous case.

D. Triangle CDE

For the intermediate triangle the NTV are v_C, v_D, and v_E, which can be generated by the combinations:

• (v_aH,v_oL) and (v_oH,v_aL), forming v_C,

• (v_bH,v_oL) and (v_oH,v_bL), forming v_D,

• (v_aH,v_bL) and (v_bH,v_aL), forming v . _E

The following three conditions define the triangle CDE:

S

The presence of three simultaneous conditions (19)-(21) makes the case of the intermediate triangle CDE the most complex among the four considered cases.

The proposed switching sequence is shown in Fig. 19(a). The parameter t_x (denoted with grey) stands for a degree of freedom which determines the relative position of the switching sequence of one inverter with respect to the other (i.e., one of the sequences can be translated by the time interval tx). A similar degree of freedom also exists in the previous three cases (grey intervals in Fig. 17 and Fig. 18). Since application times are already determined, the remaining step is to choose the value for interval tx, thus completely determining the requirements for the DSP

v A Fig. 18. Proposed switching sequences for the outer triangles (a) ACE, (b) BDE.

19(a), tx has to satisfy the following constraints (from beginning towards the end of the period):

A detailed derivation of inequalities (22)-(27) is presented in the Appendix, together with the proof that a solution always exists.

Another degree of freedom is expressed by value for interval ty, which ultimately establishing the requirements for the DSP implementation. Since using symmetric common carrier for each three legs of DSP PWM unit there can be not more than one commutation in each half-period it is important to frame the switching pattern within switching period defined by the unit. This position is defined by additional degree of freedom expressed by value for interval ty, that has to satisfy two simple conditions:

• (v_aH,v_oL) ↔ v_C:

While (28) is obvious from Fig. 19(a), a detailed derivation of (29) is presented in the Appendix, together with the proof that a solution always exists. As a value for t_y can be chosen:

2

Similarly, for the B-half the solution is analogously developed analyzing the figure Fig. 19(b):

oH

aH

with a possible solution:

2

Due to the symmetry all regarding ty can be analogously repeated for the B-half with pattern shown in Fig. 19(b) provided by swapping pairs tbH↔taH and tbL↔taL (Appendix). The solution can be:

The proposed switching sequence is discontinuous, as in previous cases. However in contrast to previous patterns contain a simultaneous commutation of two legs for both inverters H and L.

It is possible to overcome this drawback by introducing the additional vectors v_cH, v_cL and/or vdH, v_dL (depicted in Fig. 6) in the space vector decomposition (Fig. 4), leading to a more complex modulation algorithm. Despite of the asymmetric distribution of pulses within the switching period, the proposed modulation can be implemented in the PWM generation unit of an industrial DSP, as proved by the experimental tests.